transmuted sushila distribution and its application to
TRANSCRIPT
Journal of Mathematical Analysis and Modeling J Math Anal amp Modeljmamsabapubcom (2021)1(2) 1-14
ISSN 2709-5924 doi1048185jmamv1i2127
Transmuted Sushila Distribution and its Application to LifetimeData
ADETUNJI ADEMOLA A a
a Department of Statistics Federal Polytechnic Ile-Oluji Nigeria
bull Received 21 January 2021 bull Accepted 16 February 2021 bull Published Online 28 March 2021
Abstract
The Sushila distribution is generalized in this article using the quadratic rank transmutation map asdeveloped by Shaw and Buckley (2007) The newly developed distribution is called the Transmuted Sushiladistribution (TSD) Various mathematical properties of the distribution are obtained Real lifetime data isused to compare the performance of the new distribution with other related distributions The results shownby the new distribution perform creditably well
Keywords Sushila distribution Lindley distribution lifetime distribution reliability function2010 MSC 60E05 62E10 62G30 62H10 97K50
1 Introduction
Last few years have witnessed generalizations of various lifetime distributions Thisis achieved by compounding the distribution with any of new families of distributionsThe process involves introduction of new shape parameter(s) to improve flexibility ofthe baseline distribution Among well-known generalized families of distributions areMarshall-Olkin family of distributions [1] Beta G distributions [2] Quadratic Transmutedfamily of distributions [3] Kumaraswamy G distributions [4] Gamma G distributions [5]Exponentiated generalized G distributions [6] Weibull G distributions [7] and AlphaPower Transformation [8] Researchers in sciences and engineering have applied thesefamilies of distribution to improve modelling of various lifetime data
In this article we generalize the Sushila distribution [9] using the Quadratic Trans-muted family of distributions [3] and the new generalization is called the TransmutedSushila Distribution (TSD) A random variable X is said to have the Sushila distribution ifits distribution function (CDF) is given as
G(x) = 1 minusλ (1 + θ) + θ x
λ (1 + θ)eminus
θxλ (11)
lowastCorresponding author adeadetunjifedpoleledung ccopy 2020 SABA All Rights Reserved
Adetunji AA Transmuted Sushila Distribution 2
It is observed that is a special case of the Lindley distribution [10] when λ = 1
Given a baseline distribution with the CDF G(x) the Quadratic Transmuted (QT) fam-ily of distributions has the cdf
F(x) = (1 + a)G(x) minus a(G(x)
)2
(12)
QT was applied to some probability distributions by [11 12] with the resulting dis-tributions offering more flexibility [13] also discussed some mathematical properties forthe QT family of distributions With generally acceptability researchers have applied (12)and introduced different new members of the QT family for diverse lifetime distributionsList of some QT distributions was provided by [14]
2 Transmuted Sushila Distribution
The Transmuted Sushila Distribution (TSD) is obtained by putting (11) into (12)Hence a random variable X is said to have the TSD ie X sim TSD(θ λ a) if its CDF isgiven as
F(x) = (1 + a)
(1 minus
λ (1 + θ) + θ x
λ (1 + θ)eminus
θxλ
)minus a
(1 minus
λ (1 + θ) + θ x
λ (1 + θ)eminus
θxλ
)2
(21)
Figures 1 illustrates the cdf of the TSD for some selected values of scale and shapes pa-rameters The probability distribution function (PDF) of TSD is obtained by differentiating(21) once Hence the PDF of X sim TSD(θ λ a) is
f(x) =θ2
λ3(1 + θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ (22)
Note
(i) The TSD becomes the Sushila Distribution due to [9] if a = 0(ii) The TSD becomes the Lindley Distribution due to [10] if a = 0 and λ = 1The pdf of the TSD for some selected values of scale and shapes parameters is shown infigure 2
Adetunji AA Transmuted Sushila Distribution 3
Figure 1 CDF of Transmuted Sushila Distribution
Figure 2 PDF of Transmuted Sushila Distribution
3 Reliability Analysis
Survival Function The probability of an item not failing prior to a particular time isdefined by its reliability or survival function S(x) This is defined by S(x) = 1 minus F(x)Therefore if a random variable X sim TSD(θ λ a) then its survival function is given by
S(x) = 1 minus (1 + a)
(1 minus
λ (1 + θ) + θ x
λ (1 + θ)eminus
θxλ
)+ a
(1 minus
λ (1 + θ) + θ x
λ (1 + θ)eminus
θxλ
)2
(31)
Hazard Rate Function (HRF) This is the risk a system has in experiencing and event inan instantaneous time provided it has not experienced it at present time It is a measureof proneness to an event and it is obtained as h(x) =
f(x)S(x) For a random variable that
has the TSD the hrf is given as
Adetunji AA Transmuted Sushila Distribution 4
h(x) =
θ2(x+ λ)
(2a expminusθx
λ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1))
λ(λθ+ θx+ λ)((λθ+ θx)a expminusθx
λ
)minus λ(1 + θ)(aminus 1)
(32)
Figure 4 shows the HRF of the TSD for some values of scale and shapes parameters
Figure 3 The Survival function of Transmuted Sushila Distribution
Figure 4 HRF of Transmuted Sushila Distribution
Adetunji AA Transmuted Sushila Distribution 5
Figure 5 Cumulative HRF of Transmuted Sushila Distribution
Cumulative Hazard Rate Function H(x) This is the overall risk rate from the onset upto a given time The measure is obtained as H(x) = minusln(S(x))
H(x) = minusln
((λθ+ θx+ λ)
(a expminusθx
λ (λθ+ θx+ λ) minus aθλ+ θλ+ λλ2(1 + θ)2
)expminusθx
λ
) (33)
The trend depicting the cumulative HRF of the TSD for some values of scale and shapesparameters is shown in figure 5
4 Mathematical Properties
41 Order StatisticsGiven that X1n lt X2n lt lt Xnn is a set of ordered random variable of size n if
X sim TSD(θ λ a) then the PDF of the rth order statistics is given as
frn(x) =n
(rminus 1)(nminus r)f(x)[F(x)]rminus1[1 minus F(x)]nminusr
Therefore the rth order statistics of X is given as
frn(x) =nθ2(λ+ x)
(rminus 1)(nminus r)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][(1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)minus a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]rminus1
[1 minus (1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)+ a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminusr
(41)
Adetunji AA Transmuted Sushila Distribution 6
If r = 1 and r = n the 1st and nth order statistic for X are respectively given in (42)and (43)
f1n(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][
1 minus (1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)+ a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminusr
(42)
fnn(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][(1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)minus a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminus1
(43)
42 Quantiles FunctionThe quantile function of a random variable X sim TSD(θ λ a) is given as
Q(u) = minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + 2a+ a2 minus 4ua)2a
)+ 1 + θ
] (44)
Therefore the first the second and the third quartiles for the random variable are ob-tained by respectively setting u to 025 050 and 075 These are given by
Q( 14 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 + a)
2a
)+ 1 + θ
]
Q( 12 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2)
2a
)+ 1 + θ
]
Q( 34 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 minus a)
2a
)+ 1 + θ
]
where the Lambert function W is a complex function with multiple values which is definedas the solution for the equation W(u)e
W(u) = u
Equation (44) is obtained using Maple 2016 [15]
Adetunji AA Transmuted Sushila Distribution 7
43 Skewness and KurtosisVariability in a data set can be investigated using skewness and kurtosis and classical
measures can be susceptible to outliers Given the quantile function as in (44) the MoorrsquosKurtosis [16] based on octiles is given as
KM =Q 7
8minusQ 5
8+Q 3
8minusQ 1
8
Q 68minusQ 2
8
(45)
Also the Bowleyrsquos measure of skewness [17] based on quartiles is given as
SKB =Q 3
4minus 2Q 2
4+Q 1
4
Q 34minusQ 1
4
(46)
44 MomentsProposition 41 If a random variable X follows TSD with pdf as given in (22) then thekth raw moment is given by
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1
minus (aθ+ aminus θminus 1)(θ+ k+ 1) +a(k+ 1)(k+ 2)
2k+2
) (47)
Proof
micro1k = E(xk) =
intinfin0xkf(x)dx
=
intinfin0xk
θ2
λ3(1 + θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(λθ2aeminus
θxλ
+θx2aeminusθxλ + λ2aeminus
θxλ minus aθλminus aλ+ θλ+ λ
)dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ+ 2aλ
)P2
minusλ2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0xkeminus
2θxλ dx =
λk+1k2k+1θk+1 P2 =
intinfin0xk+1eminus
2θxλ dx =
λk+2(k+ 1)2k+2θk+2
Adetunji AA Transmuted Sushila Distribution 8
P3 =
intinfin0xkeminus
θxλ dx =
λk+1kθk+1 P4 =
intinfin0xk+2eminus
2θxλ dx =
λk+3k2k+3θk+2
P5 =
intinfin0xk+1eminus
θxλ dx =
λk+2(k+ 1)θk+2
Therefore
E(xk) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)(λk+1k
2k+1θk+1
)+
(4aθλ+ 2aλ
)(λk+2(k+ 1)
2k+2θk+2
)minusλ2
(aθ+ aminus θminus 1
)(λk+1kθk+1
)+ 2aθ
(λk+3k
2k+3θk+2
)minusλ
(aθ+ aminus θminus 1
)(λk+2(k+ 1)
θk+2
)
Hence
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1 minus(aθ+aminusθminus1)(θ+k+1)+a(k+ 1)(k+ 2)
2k+2
)
Mean and Variance of TSD The mean of a random variable X follows TSD is obtainedby putting k = 1 in (47) above Hence the mean is given as
E(x) =λ
4θ(1 + θ)2
(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8
)Also the variance of X is obtained as var(x) = E(x2) minus (E(x))2 where E(x2) is obtained byequating k = 2 in (47)
E(x2) =λ2
4θ2(1 + θ)2
(6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24
)Hence the variance of X is given by
var(x) =λ2
4θ2(1 + θ)2
[(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8)2
4(1 + θ)2
+ (6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24)]
Adetunji AA Transmuted Sushila Distribution 9
45 Moment Generating FunctionProposition 42 The moment generating function of a random variable X that followsthe TSD is given as
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3
minus(aθ+ aminus θminus 1)(θminus λt+ 1)
(θminus λt)2
)
(48)
Proof
Mx(t) = E(etx) =
intinfin0etxf(x)dx
=
intinfin0etx
θ2
λ3(1 + θ)2 (λ+ x)
(2aeminus
θxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0(λ+ x)eminus
θminusλtλ x
(2aθλeminus
θxλ + 2aθxeminus
θxλ + 2aλeminus
θxλ
minusλ(aθ+ aminus θminus 1))dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ
+2aλ)P2 minus λ
2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0eminus
2θminusλtλ x dx =
λ
2θminus λt P2 =
intinfin0xeminus
2θminusλtλ x dx =
λ2
(2θminus λt)2
P3 =
intinfin0
eminusθminusλtλ x dx =
λ
(θminus λt) P4 =
intinfin0
x2eminus2θminusλtλ x dx =
2λ3
(2θminus λt)3
P5 =
intinfin0
xeminusθminusλtλ x dx =
λ2
(θminus λt)2
Adetunji AA Transmuted Sushila Distribution 10
Therefore
E(etx) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2)
(λ
2θminus λt
)+ (4aθλ+ 2aλ)
(λ
(2θminus λt)
)2
minusλ2(aθ+ aminus θminus 1)(
λ
θminus λt
)+ 4aθ
(2λ
(2θminus λt)
)3
minus λ(aθ+ aminus θminus 1)(
λ
(θminus λt)
=
(θ2
λ3(1 + θ)2
)[2aλ2(1 + θ)
(λ
2θminus λt
)+2aλ(1 + 2θ)
(λ
2θminus λt
)2
+ 4aθ(
λ
2θminus λt
)3
minus(aθ+ aminus θminus 1)λ3(
1 +1
θminus λt
)]=
(θ2
λ3(1 + θ)2
)[2a(1 + θ)
(1
2θminus λt
)+2a(1 + 2θ)
(1
2θminus λt
)2
+ 4aθ(
12θminus λt
)2
minus(aθ+ aminus θminus 1)λ3(θminus λt+ 1θminus λt
)]
Hence
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3 minus
(aθ+ aminus θminus 1)(θminus λt+ 1)(θminus λt)2
)
5 Parameter Estimation
Proposition 51 Given that Xi i = 1 2 n are iid random variables from TSD then thelog-likelihood function of X is defined as
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
(51)
Adetunji AA Transmuted Sushila Distribution 11
Proof The likelihood function of a random variable X that follows TSD is
L =
i=nprodi=1
(θ2
λ3(1 + θ)2
)(λ+ xi)
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλxi
=
(θ2n
λ3n(1 + θ)2n
) i=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
=
(θ
1 + θ
)2n
λminus3ni=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
Hence the log-likelihood function of a random variable X that follows TSD is
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
The MLE of (θ λa) can be obtained by maximizing (51) This gives the set of normalequations below
partlogL
partθ=
2nθ
minus2n
1 + θminus
1λ
nsumi=1
xi
+
nsumi=1
minus 1λ2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(λ+ xi + λ) minus aλ+ λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
partlogL
partλ=
minus3nθ
+
nsumi=1
1λ+ xi
+θ
λ2
nsumi=1
xi
+
nsumi=1
minus 1λ2 2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(1 + θ) minus aθminus a+ θ+ 1
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
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Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
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[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 2
It is observed that is a special case of the Lindley distribution [10] when λ = 1
Given a baseline distribution with the CDF G(x) the Quadratic Transmuted (QT) fam-ily of distributions has the cdf
F(x) = (1 + a)G(x) minus a(G(x)
)2
(12)
QT was applied to some probability distributions by [11 12] with the resulting dis-tributions offering more flexibility [13] also discussed some mathematical properties forthe QT family of distributions With generally acceptability researchers have applied (12)and introduced different new members of the QT family for diverse lifetime distributionsList of some QT distributions was provided by [14]
2 Transmuted Sushila Distribution
The Transmuted Sushila Distribution (TSD) is obtained by putting (11) into (12)Hence a random variable X is said to have the TSD ie X sim TSD(θ λ a) if its CDF isgiven as
F(x) = (1 + a)
(1 minus
λ (1 + θ) + θ x
λ (1 + θ)eminus
θxλ
)minus a
(1 minus
λ (1 + θ) + θ x
λ (1 + θ)eminus
θxλ
)2
(21)
Figures 1 illustrates the cdf of the TSD for some selected values of scale and shapes pa-rameters The probability distribution function (PDF) of TSD is obtained by differentiating(21) once Hence the PDF of X sim TSD(θ λ a) is
f(x) =θ2
λ3(1 + θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ (22)
Note
(i) The TSD becomes the Sushila Distribution due to [9] if a = 0(ii) The TSD becomes the Lindley Distribution due to [10] if a = 0 and λ = 1The pdf of the TSD for some selected values of scale and shapes parameters is shown infigure 2
Adetunji AA Transmuted Sushila Distribution 3
Figure 1 CDF of Transmuted Sushila Distribution
Figure 2 PDF of Transmuted Sushila Distribution
3 Reliability Analysis
Survival Function The probability of an item not failing prior to a particular time isdefined by its reliability or survival function S(x) This is defined by S(x) = 1 minus F(x)Therefore if a random variable X sim TSD(θ λ a) then its survival function is given by
S(x) = 1 minus (1 + a)
(1 minus
λ (1 + θ) + θ x
λ (1 + θ)eminus
θxλ
)+ a
(1 minus
λ (1 + θ) + θ x
λ (1 + θ)eminus
θxλ
)2
(31)
Hazard Rate Function (HRF) This is the risk a system has in experiencing and event inan instantaneous time provided it has not experienced it at present time It is a measureof proneness to an event and it is obtained as h(x) =
f(x)S(x) For a random variable that
has the TSD the hrf is given as
Adetunji AA Transmuted Sushila Distribution 4
h(x) =
θ2(x+ λ)
(2a expminusθx
λ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1))
λ(λθ+ θx+ λ)((λθ+ θx)a expminusθx
λ
)minus λ(1 + θ)(aminus 1)
(32)
Figure 4 shows the HRF of the TSD for some values of scale and shapes parameters
Figure 3 The Survival function of Transmuted Sushila Distribution
Figure 4 HRF of Transmuted Sushila Distribution
Adetunji AA Transmuted Sushila Distribution 5
Figure 5 Cumulative HRF of Transmuted Sushila Distribution
Cumulative Hazard Rate Function H(x) This is the overall risk rate from the onset upto a given time The measure is obtained as H(x) = minusln(S(x))
H(x) = minusln
((λθ+ θx+ λ)
(a expminusθx
λ (λθ+ θx+ λ) minus aθλ+ θλ+ λλ2(1 + θ)2
)expminusθx
λ
) (33)
The trend depicting the cumulative HRF of the TSD for some values of scale and shapesparameters is shown in figure 5
4 Mathematical Properties
41 Order StatisticsGiven that X1n lt X2n lt lt Xnn is a set of ordered random variable of size n if
X sim TSD(θ λ a) then the PDF of the rth order statistics is given as
frn(x) =n
(rminus 1)(nminus r)f(x)[F(x)]rminus1[1 minus F(x)]nminusr
Therefore the rth order statistics of X is given as
frn(x) =nθ2(λ+ x)
(rminus 1)(nminus r)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][(1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)minus a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]rminus1
[1 minus (1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)+ a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminusr
(41)
Adetunji AA Transmuted Sushila Distribution 6
If r = 1 and r = n the 1st and nth order statistic for X are respectively given in (42)and (43)
f1n(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][
1 minus (1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)+ a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminusr
(42)
fnn(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][(1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)minus a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminus1
(43)
42 Quantiles FunctionThe quantile function of a random variable X sim TSD(θ λ a) is given as
Q(u) = minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + 2a+ a2 minus 4ua)2a
)+ 1 + θ
] (44)
Therefore the first the second and the third quartiles for the random variable are ob-tained by respectively setting u to 025 050 and 075 These are given by
Q( 14 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 + a)
2a
)+ 1 + θ
]
Q( 12 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2)
2a
)+ 1 + θ
]
Q( 34 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 minus a)
2a
)+ 1 + θ
]
where the Lambert function W is a complex function with multiple values which is definedas the solution for the equation W(u)e
W(u) = u
Equation (44) is obtained using Maple 2016 [15]
Adetunji AA Transmuted Sushila Distribution 7
43 Skewness and KurtosisVariability in a data set can be investigated using skewness and kurtosis and classical
measures can be susceptible to outliers Given the quantile function as in (44) the MoorrsquosKurtosis [16] based on octiles is given as
KM =Q 7
8minusQ 5
8+Q 3
8minusQ 1
8
Q 68minusQ 2
8
(45)
Also the Bowleyrsquos measure of skewness [17] based on quartiles is given as
SKB =Q 3
4minus 2Q 2
4+Q 1
4
Q 34minusQ 1
4
(46)
44 MomentsProposition 41 If a random variable X follows TSD with pdf as given in (22) then thekth raw moment is given by
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1
minus (aθ+ aminus θminus 1)(θ+ k+ 1) +a(k+ 1)(k+ 2)
2k+2
) (47)
Proof
micro1k = E(xk) =
intinfin0xkf(x)dx
=
intinfin0xk
θ2
λ3(1 + θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(λθ2aeminus
θxλ
+θx2aeminusθxλ + λ2aeminus
θxλ minus aθλminus aλ+ θλ+ λ
)dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ+ 2aλ
)P2
minusλ2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0xkeminus
2θxλ dx =
λk+1k2k+1θk+1 P2 =
intinfin0xk+1eminus
2θxλ dx =
λk+2(k+ 1)2k+2θk+2
Adetunji AA Transmuted Sushila Distribution 8
P3 =
intinfin0xkeminus
θxλ dx =
λk+1kθk+1 P4 =
intinfin0xk+2eminus
2θxλ dx =
λk+3k2k+3θk+2
P5 =
intinfin0xk+1eminus
θxλ dx =
λk+2(k+ 1)θk+2
Therefore
E(xk) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)(λk+1k
2k+1θk+1
)+
(4aθλ+ 2aλ
)(λk+2(k+ 1)
2k+2θk+2
)minusλ2
(aθ+ aminus θminus 1
)(λk+1kθk+1
)+ 2aθ
(λk+3k
2k+3θk+2
)minusλ
(aθ+ aminus θminus 1
)(λk+2(k+ 1)
θk+2
)
Hence
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1 minus(aθ+aminusθminus1)(θ+k+1)+a(k+ 1)(k+ 2)
2k+2
)
Mean and Variance of TSD The mean of a random variable X follows TSD is obtainedby putting k = 1 in (47) above Hence the mean is given as
E(x) =λ
4θ(1 + θ)2
(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8
)Also the variance of X is obtained as var(x) = E(x2) minus (E(x))2 where E(x2) is obtained byequating k = 2 in (47)
E(x2) =λ2
4θ2(1 + θ)2
(6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24
)Hence the variance of X is given by
var(x) =λ2
4θ2(1 + θ)2
[(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8)2
4(1 + θ)2
+ (6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24)]
Adetunji AA Transmuted Sushila Distribution 9
45 Moment Generating FunctionProposition 42 The moment generating function of a random variable X that followsthe TSD is given as
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3
minus(aθ+ aminus θminus 1)(θminus λt+ 1)
(θminus λt)2
)
(48)
Proof
Mx(t) = E(etx) =
intinfin0etxf(x)dx
=
intinfin0etx
θ2
λ3(1 + θ)2 (λ+ x)
(2aeminus
θxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0(λ+ x)eminus
θminusλtλ x
(2aθλeminus
θxλ + 2aθxeminus
θxλ + 2aλeminus
θxλ
minusλ(aθ+ aminus θminus 1))dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ
+2aλ)P2 minus λ
2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0eminus
2θminusλtλ x dx =
λ
2θminus λt P2 =
intinfin0xeminus
2θminusλtλ x dx =
λ2
(2θminus λt)2
P3 =
intinfin0
eminusθminusλtλ x dx =
λ
(θminus λt) P4 =
intinfin0
x2eminus2θminusλtλ x dx =
2λ3
(2θminus λt)3
P5 =
intinfin0
xeminusθminusλtλ x dx =
λ2
(θminus λt)2
Adetunji AA Transmuted Sushila Distribution 10
Therefore
E(etx) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2)
(λ
2θminus λt
)+ (4aθλ+ 2aλ)
(λ
(2θminus λt)
)2
minusλ2(aθ+ aminus θminus 1)(
λ
θminus λt
)+ 4aθ
(2λ
(2θminus λt)
)3
minus λ(aθ+ aminus θminus 1)(
λ
(θminus λt)
=
(θ2
λ3(1 + θ)2
)[2aλ2(1 + θ)
(λ
2θminus λt
)+2aλ(1 + 2θ)
(λ
2θminus λt
)2
+ 4aθ(
λ
2θminus λt
)3
minus(aθ+ aminus θminus 1)λ3(
1 +1
θminus λt
)]=
(θ2
λ3(1 + θ)2
)[2a(1 + θ)
(1
2θminus λt
)+2a(1 + 2θ)
(1
2θminus λt
)2
+ 4aθ(
12θminus λt
)2
minus(aθ+ aminus θminus 1)λ3(θminus λt+ 1θminus λt
)]
Hence
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3 minus
(aθ+ aminus θminus 1)(θminus λt+ 1)(θminus λt)2
)
5 Parameter Estimation
Proposition 51 Given that Xi i = 1 2 n are iid random variables from TSD then thelog-likelihood function of X is defined as
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
(51)
Adetunji AA Transmuted Sushila Distribution 11
Proof The likelihood function of a random variable X that follows TSD is
L =
i=nprodi=1
(θ2
λ3(1 + θ)2
)(λ+ xi)
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλxi
=
(θ2n
λ3n(1 + θ)2n
) i=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
=
(θ
1 + θ
)2n
λminus3ni=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
Hence the log-likelihood function of a random variable X that follows TSD is
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
The MLE of (θ λa) can be obtained by maximizing (51) This gives the set of normalequations below
partlogL
partθ=
2nθ
minus2n
1 + θminus
1λ
nsumi=1
xi
+
nsumi=1
minus 1λ2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(λ+ xi + λ) minus aλ+ λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
partlogL
partλ=
minus3nθ
+
nsumi=1
1λ+ xi
+θ
λ2
nsumi=1
xi
+
nsumi=1
minus 1λ2 2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(1 + θ) minus aθminus a+ θ+ 1
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 3
Figure 1 CDF of Transmuted Sushila Distribution
Figure 2 PDF of Transmuted Sushila Distribution
3 Reliability Analysis
Survival Function The probability of an item not failing prior to a particular time isdefined by its reliability or survival function S(x) This is defined by S(x) = 1 minus F(x)Therefore if a random variable X sim TSD(θ λ a) then its survival function is given by
S(x) = 1 minus (1 + a)
(1 minus
λ (1 + θ) + θ x
λ (1 + θ)eminus
θxλ
)+ a
(1 minus
λ (1 + θ) + θ x
λ (1 + θ)eminus
θxλ
)2
(31)
Hazard Rate Function (HRF) This is the risk a system has in experiencing and event inan instantaneous time provided it has not experienced it at present time It is a measureof proneness to an event and it is obtained as h(x) =
f(x)S(x) For a random variable that
has the TSD the hrf is given as
Adetunji AA Transmuted Sushila Distribution 4
h(x) =
θ2(x+ λ)
(2a expminusθx
λ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1))
λ(λθ+ θx+ λ)((λθ+ θx)a expminusθx
λ
)minus λ(1 + θ)(aminus 1)
(32)
Figure 4 shows the HRF of the TSD for some values of scale and shapes parameters
Figure 3 The Survival function of Transmuted Sushila Distribution
Figure 4 HRF of Transmuted Sushila Distribution
Adetunji AA Transmuted Sushila Distribution 5
Figure 5 Cumulative HRF of Transmuted Sushila Distribution
Cumulative Hazard Rate Function H(x) This is the overall risk rate from the onset upto a given time The measure is obtained as H(x) = minusln(S(x))
H(x) = minusln
((λθ+ θx+ λ)
(a expminusθx
λ (λθ+ θx+ λ) minus aθλ+ θλ+ λλ2(1 + θ)2
)expminusθx
λ
) (33)
The trend depicting the cumulative HRF of the TSD for some values of scale and shapesparameters is shown in figure 5
4 Mathematical Properties
41 Order StatisticsGiven that X1n lt X2n lt lt Xnn is a set of ordered random variable of size n if
X sim TSD(θ λ a) then the PDF of the rth order statistics is given as
frn(x) =n
(rminus 1)(nminus r)f(x)[F(x)]rminus1[1 minus F(x)]nminusr
Therefore the rth order statistics of X is given as
frn(x) =nθ2(λ+ x)
(rminus 1)(nminus r)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][(1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)minus a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]rminus1
[1 minus (1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)+ a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminusr
(41)
Adetunji AA Transmuted Sushila Distribution 6
If r = 1 and r = n the 1st and nth order statistic for X are respectively given in (42)and (43)
f1n(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][
1 minus (1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)+ a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminusr
(42)
fnn(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][(1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)minus a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminus1
(43)
42 Quantiles FunctionThe quantile function of a random variable X sim TSD(θ λ a) is given as
Q(u) = minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + 2a+ a2 minus 4ua)2a
)+ 1 + θ
] (44)
Therefore the first the second and the third quartiles for the random variable are ob-tained by respectively setting u to 025 050 and 075 These are given by
Q( 14 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 + a)
2a
)+ 1 + θ
]
Q( 12 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2)
2a
)+ 1 + θ
]
Q( 34 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 minus a)
2a
)+ 1 + θ
]
where the Lambert function W is a complex function with multiple values which is definedas the solution for the equation W(u)e
W(u) = u
Equation (44) is obtained using Maple 2016 [15]
Adetunji AA Transmuted Sushila Distribution 7
43 Skewness and KurtosisVariability in a data set can be investigated using skewness and kurtosis and classical
measures can be susceptible to outliers Given the quantile function as in (44) the MoorrsquosKurtosis [16] based on octiles is given as
KM =Q 7
8minusQ 5
8+Q 3
8minusQ 1
8
Q 68minusQ 2
8
(45)
Also the Bowleyrsquos measure of skewness [17] based on quartiles is given as
SKB =Q 3
4minus 2Q 2
4+Q 1
4
Q 34minusQ 1
4
(46)
44 MomentsProposition 41 If a random variable X follows TSD with pdf as given in (22) then thekth raw moment is given by
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1
minus (aθ+ aminus θminus 1)(θ+ k+ 1) +a(k+ 1)(k+ 2)
2k+2
) (47)
Proof
micro1k = E(xk) =
intinfin0xkf(x)dx
=
intinfin0xk
θ2
λ3(1 + θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(λθ2aeminus
θxλ
+θx2aeminusθxλ + λ2aeminus
θxλ minus aθλminus aλ+ θλ+ λ
)dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ+ 2aλ
)P2
minusλ2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0xkeminus
2θxλ dx =
λk+1k2k+1θk+1 P2 =
intinfin0xk+1eminus
2θxλ dx =
λk+2(k+ 1)2k+2θk+2
Adetunji AA Transmuted Sushila Distribution 8
P3 =
intinfin0xkeminus
θxλ dx =
λk+1kθk+1 P4 =
intinfin0xk+2eminus
2θxλ dx =
λk+3k2k+3θk+2
P5 =
intinfin0xk+1eminus
θxλ dx =
λk+2(k+ 1)θk+2
Therefore
E(xk) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)(λk+1k
2k+1θk+1
)+
(4aθλ+ 2aλ
)(λk+2(k+ 1)
2k+2θk+2
)minusλ2
(aθ+ aminus θminus 1
)(λk+1kθk+1
)+ 2aθ
(λk+3k
2k+3θk+2
)minusλ
(aθ+ aminus θminus 1
)(λk+2(k+ 1)
θk+2
)
Hence
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1 minus(aθ+aminusθminus1)(θ+k+1)+a(k+ 1)(k+ 2)
2k+2
)
Mean and Variance of TSD The mean of a random variable X follows TSD is obtainedby putting k = 1 in (47) above Hence the mean is given as
E(x) =λ
4θ(1 + θ)2
(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8
)Also the variance of X is obtained as var(x) = E(x2) minus (E(x))2 where E(x2) is obtained byequating k = 2 in (47)
E(x2) =λ2
4θ2(1 + θ)2
(6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24
)Hence the variance of X is given by
var(x) =λ2
4θ2(1 + θ)2
[(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8)2
4(1 + θ)2
+ (6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24)]
Adetunji AA Transmuted Sushila Distribution 9
45 Moment Generating FunctionProposition 42 The moment generating function of a random variable X that followsthe TSD is given as
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3
minus(aθ+ aminus θminus 1)(θminus λt+ 1)
(θminus λt)2
)
(48)
Proof
Mx(t) = E(etx) =
intinfin0etxf(x)dx
=
intinfin0etx
θ2
λ3(1 + θ)2 (λ+ x)
(2aeminus
θxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0(λ+ x)eminus
θminusλtλ x
(2aθλeminus
θxλ + 2aθxeminus
θxλ + 2aλeminus
θxλ
minusλ(aθ+ aminus θminus 1))dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ
+2aλ)P2 minus λ
2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0eminus
2θminusλtλ x dx =
λ
2θminus λt P2 =
intinfin0xeminus
2θminusλtλ x dx =
λ2
(2θminus λt)2
P3 =
intinfin0
eminusθminusλtλ x dx =
λ
(θminus λt) P4 =
intinfin0
x2eminus2θminusλtλ x dx =
2λ3
(2θminus λt)3
P5 =
intinfin0
xeminusθminusλtλ x dx =
λ2
(θminus λt)2
Adetunji AA Transmuted Sushila Distribution 10
Therefore
E(etx) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2)
(λ
2θminus λt
)+ (4aθλ+ 2aλ)
(λ
(2θminus λt)
)2
minusλ2(aθ+ aminus θminus 1)(
λ
θminus λt
)+ 4aθ
(2λ
(2θminus λt)
)3
minus λ(aθ+ aminus θminus 1)(
λ
(θminus λt)
=
(θ2
λ3(1 + θ)2
)[2aλ2(1 + θ)
(λ
2θminus λt
)+2aλ(1 + 2θ)
(λ
2θminus λt
)2
+ 4aθ(
λ
2θminus λt
)3
minus(aθ+ aminus θminus 1)λ3(
1 +1
θminus λt
)]=
(θ2
λ3(1 + θ)2
)[2a(1 + θ)
(1
2θminus λt
)+2a(1 + 2θ)
(1
2θminus λt
)2
+ 4aθ(
12θminus λt
)2
minus(aθ+ aminus θminus 1)λ3(θminus λt+ 1θminus λt
)]
Hence
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3 minus
(aθ+ aminus θminus 1)(θminus λt+ 1)(θminus λt)2
)
5 Parameter Estimation
Proposition 51 Given that Xi i = 1 2 n are iid random variables from TSD then thelog-likelihood function of X is defined as
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
(51)
Adetunji AA Transmuted Sushila Distribution 11
Proof The likelihood function of a random variable X that follows TSD is
L =
i=nprodi=1
(θ2
λ3(1 + θ)2
)(λ+ xi)
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλxi
=
(θ2n
λ3n(1 + θ)2n
) i=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
=
(θ
1 + θ
)2n
λminus3ni=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
Hence the log-likelihood function of a random variable X that follows TSD is
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
The MLE of (θ λa) can be obtained by maximizing (51) This gives the set of normalequations below
partlogL
partθ=
2nθ
minus2n
1 + θminus
1λ
nsumi=1
xi
+
nsumi=1
minus 1λ2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(λ+ xi + λ) minus aλ+ λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
partlogL
partλ=
minus3nθ
+
nsumi=1
1λ+ xi
+θ
λ2
nsumi=1
xi
+
nsumi=1
minus 1λ2 2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(1 + θ) minus aθminus a+ θ+ 1
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 4
h(x) =
θ2(x+ λ)
(2a expminusθx
λ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1))
λ(λθ+ θx+ λ)((λθ+ θx)a expminusθx
λ
)minus λ(1 + θ)(aminus 1)
(32)
Figure 4 shows the HRF of the TSD for some values of scale and shapes parameters
Figure 3 The Survival function of Transmuted Sushila Distribution
Figure 4 HRF of Transmuted Sushila Distribution
Adetunji AA Transmuted Sushila Distribution 5
Figure 5 Cumulative HRF of Transmuted Sushila Distribution
Cumulative Hazard Rate Function H(x) This is the overall risk rate from the onset upto a given time The measure is obtained as H(x) = minusln(S(x))
H(x) = minusln
((λθ+ θx+ λ)
(a expminusθx
λ (λθ+ θx+ λ) minus aθλ+ θλ+ λλ2(1 + θ)2
)expminusθx
λ
) (33)
The trend depicting the cumulative HRF of the TSD for some values of scale and shapesparameters is shown in figure 5
4 Mathematical Properties
41 Order StatisticsGiven that X1n lt X2n lt lt Xnn is a set of ordered random variable of size n if
X sim TSD(θ λ a) then the PDF of the rth order statistics is given as
frn(x) =n
(rminus 1)(nminus r)f(x)[F(x)]rminus1[1 minus F(x)]nminusr
Therefore the rth order statistics of X is given as
frn(x) =nθ2(λ+ x)
(rminus 1)(nminus r)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][(1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)minus a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]rminus1
[1 minus (1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)+ a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminusr
(41)
Adetunji AA Transmuted Sushila Distribution 6
If r = 1 and r = n the 1st and nth order statistic for X are respectively given in (42)and (43)
f1n(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][
1 minus (1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)+ a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminusr
(42)
fnn(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][(1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)minus a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminus1
(43)
42 Quantiles FunctionThe quantile function of a random variable X sim TSD(θ λ a) is given as
Q(u) = minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + 2a+ a2 minus 4ua)2a
)+ 1 + θ
] (44)
Therefore the first the second and the third quartiles for the random variable are ob-tained by respectively setting u to 025 050 and 075 These are given by
Q( 14 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 + a)
2a
)+ 1 + θ
]
Q( 12 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2)
2a
)+ 1 + θ
]
Q( 34 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 minus a)
2a
)+ 1 + θ
]
where the Lambert function W is a complex function with multiple values which is definedas the solution for the equation W(u)e
W(u) = u
Equation (44) is obtained using Maple 2016 [15]
Adetunji AA Transmuted Sushila Distribution 7
43 Skewness and KurtosisVariability in a data set can be investigated using skewness and kurtosis and classical
measures can be susceptible to outliers Given the quantile function as in (44) the MoorrsquosKurtosis [16] based on octiles is given as
KM =Q 7
8minusQ 5
8+Q 3
8minusQ 1
8
Q 68minusQ 2
8
(45)
Also the Bowleyrsquos measure of skewness [17] based on quartiles is given as
SKB =Q 3
4minus 2Q 2
4+Q 1
4
Q 34minusQ 1
4
(46)
44 MomentsProposition 41 If a random variable X follows TSD with pdf as given in (22) then thekth raw moment is given by
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1
minus (aθ+ aminus θminus 1)(θ+ k+ 1) +a(k+ 1)(k+ 2)
2k+2
) (47)
Proof
micro1k = E(xk) =
intinfin0xkf(x)dx
=
intinfin0xk
θ2
λ3(1 + θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(λθ2aeminus
θxλ
+θx2aeminusθxλ + λ2aeminus
θxλ minus aθλminus aλ+ θλ+ λ
)dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ+ 2aλ
)P2
minusλ2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0xkeminus
2θxλ dx =
λk+1k2k+1θk+1 P2 =
intinfin0xk+1eminus
2θxλ dx =
λk+2(k+ 1)2k+2θk+2
Adetunji AA Transmuted Sushila Distribution 8
P3 =
intinfin0xkeminus
θxλ dx =
λk+1kθk+1 P4 =
intinfin0xk+2eminus
2θxλ dx =
λk+3k2k+3θk+2
P5 =
intinfin0xk+1eminus
θxλ dx =
λk+2(k+ 1)θk+2
Therefore
E(xk) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)(λk+1k
2k+1θk+1
)+
(4aθλ+ 2aλ
)(λk+2(k+ 1)
2k+2θk+2
)minusλ2
(aθ+ aminus θminus 1
)(λk+1kθk+1
)+ 2aθ
(λk+3k
2k+3θk+2
)minusλ
(aθ+ aminus θminus 1
)(λk+2(k+ 1)
θk+2
)
Hence
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1 minus(aθ+aminusθminus1)(θ+k+1)+a(k+ 1)(k+ 2)
2k+2
)
Mean and Variance of TSD The mean of a random variable X follows TSD is obtainedby putting k = 1 in (47) above Hence the mean is given as
E(x) =λ
4θ(1 + θ)2
(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8
)Also the variance of X is obtained as var(x) = E(x2) minus (E(x))2 where E(x2) is obtained byequating k = 2 in (47)
E(x2) =λ2
4θ2(1 + θ)2
(6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24
)Hence the variance of X is given by
var(x) =λ2
4θ2(1 + θ)2
[(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8)2
4(1 + θ)2
+ (6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24)]
Adetunji AA Transmuted Sushila Distribution 9
45 Moment Generating FunctionProposition 42 The moment generating function of a random variable X that followsthe TSD is given as
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3
minus(aθ+ aminus θminus 1)(θminus λt+ 1)
(θminus λt)2
)
(48)
Proof
Mx(t) = E(etx) =
intinfin0etxf(x)dx
=
intinfin0etx
θ2
λ3(1 + θ)2 (λ+ x)
(2aeminus
θxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0(λ+ x)eminus
θminusλtλ x
(2aθλeminus
θxλ + 2aθxeminus
θxλ + 2aλeminus
θxλ
minusλ(aθ+ aminus θminus 1))dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ
+2aλ)P2 minus λ
2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0eminus
2θminusλtλ x dx =
λ
2θminus λt P2 =
intinfin0xeminus
2θminusλtλ x dx =
λ2
(2θminus λt)2
P3 =
intinfin0
eminusθminusλtλ x dx =
λ
(θminus λt) P4 =
intinfin0
x2eminus2θminusλtλ x dx =
2λ3
(2θminus λt)3
P5 =
intinfin0
xeminusθminusλtλ x dx =
λ2
(θminus λt)2
Adetunji AA Transmuted Sushila Distribution 10
Therefore
E(etx) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2)
(λ
2θminus λt
)+ (4aθλ+ 2aλ)
(λ
(2θminus λt)
)2
minusλ2(aθ+ aminus θminus 1)(
λ
θminus λt
)+ 4aθ
(2λ
(2θminus λt)
)3
minus λ(aθ+ aminus θminus 1)(
λ
(θminus λt)
=
(θ2
λ3(1 + θ)2
)[2aλ2(1 + θ)
(λ
2θminus λt
)+2aλ(1 + 2θ)
(λ
2θminus λt
)2
+ 4aθ(
λ
2θminus λt
)3
minus(aθ+ aminus θminus 1)λ3(
1 +1
θminus λt
)]=
(θ2
λ3(1 + θ)2
)[2a(1 + θ)
(1
2θminus λt
)+2a(1 + 2θ)
(1
2θminus λt
)2
+ 4aθ(
12θminus λt
)2
minus(aθ+ aminus θminus 1)λ3(θminus λt+ 1θminus λt
)]
Hence
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3 minus
(aθ+ aminus θminus 1)(θminus λt+ 1)(θminus λt)2
)
5 Parameter Estimation
Proposition 51 Given that Xi i = 1 2 n are iid random variables from TSD then thelog-likelihood function of X is defined as
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
(51)
Adetunji AA Transmuted Sushila Distribution 11
Proof The likelihood function of a random variable X that follows TSD is
L =
i=nprodi=1
(θ2
λ3(1 + θ)2
)(λ+ xi)
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλxi
=
(θ2n
λ3n(1 + θ)2n
) i=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
=
(θ
1 + θ
)2n
λminus3ni=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
Hence the log-likelihood function of a random variable X that follows TSD is
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
The MLE of (θ λa) can be obtained by maximizing (51) This gives the set of normalequations below
partlogL
partθ=
2nθ
minus2n
1 + θminus
1λ
nsumi=1
xi
+
nsumi=1
minus 1λ2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(λ+ xi + λ) minus aλ+ λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
partlogL
partλ=
minus3nθ
+
nsumi=1
1λ+ xi
+θ
λ2
nsumi=1
xi
+
nsumi=1
minus 1λ2 2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(1 + θ) minus aθminus a+ θ+ 1
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 5
Figure 5 Cumulative HRF of Transmuted Sushila Distribution
Cumulative Hazard Rate Function H(x) This is the overall risk rate from the onset upto a given time The measure is obtained as H(x) = minusln(S(x))
H(x) = minusln
((λθ+ θx+ λ)
(a expminusθx
λ (λθ+ θx+ λ) minus aθλ+ θλ+ λλ2(1 + θ)2
)expminusθx
λ
) (33)
The trend depicting the cumulative HRF of the TSD for some values of scale and shapesparameters is shown in figure 5
4 Mathematical Properties
41 Order StatisticsGiven that X1n lt X2n lt lt Xnn is a set of ordered random variable of size n if
X sim TSD(θ λ a) then the PDF of the rth order statistics is given as
frn(x) =n
(rminus 1)(nminus r)f(x)[F(x)]rminus1[1 minus F(x)]nminusr
Therefore the rth order statistics of X is given as
frn(x) =nθ2(λ+ x)
(rminus 1)(nminus r)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][(1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)minus a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]rminus1
[1 minus (1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)+ a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminusr
(41)
Adetunji AA Transmuted Sushila Distribution 6
If r = 1 and r = n the 1st and nth order statistic for X are respectively given in (42)and (43)
f1n(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][
1 minus (1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)+ a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminusr
(42)
fnn(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][(1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)minus a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminus1
(43)
42 Quantiles FunctionThe quantile function of a random variable X sim TSD(θ λ a) is given as
Q(u) = minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + 2a+ a2 minus 4ua)2a
)+ 1 + θ
] (44)
Therefore the first the second and the third quartiles for the random variable are ob-tained by respectively setting u to 025 050 and 075 These are given by
Q( 14 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 + a)
2a
)+ 1 + θ
]
Q( 12 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2)
2a
)+ 1 + θ
]
Q( 34 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 minus a)
2a
)+ 1 + θ
]
where the Lambert function W is a complex function with multiple values which is definedas the solution for the equation W(u)e
W(u) = u
Equation (44) is obtained using Maple 2016 [15]
Adetunji AA Transmuted Sushila Distribution 7
43 Skewness and KurtosisVariability in a data set can be investigated using skewness and kurtosis and classical
measures can be susceptible to outliers Given the quantile function as in (44) the MoorrsquosKurtosis [16] based on octiles is given as
KM =Q 7
8minusQ 5
8+Q 3
8minusQ 1
8
Q 68minusQ 2
8
(45)
Also the Bowleyrsquos measure of skewness [17] based on quartiles is given as
SKB =Q 3
4minus 2Q 2
4+Q 1
4
Q 34minusQ 1
4
(46)
44 MomentsProposition 41 If a random variable X follows TSD with pdf as given in (22) then thekth raw moment is given by
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1
minus (aθ+ aminus θminus 1)(θ+ k+ 1) +a(k+ 1)(k+ 2)
2k+2
) (47)
Proof
micro1k = E(xk) =
intinfin0xkf(x)dx
=
intinfin0xk
θ2
λ3(1 + θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(λθ2aeminus
θxλ
+θx2aeminusθxλ + λ2aeminus
θxλ minus aθλminus aλ+ θλ+ λ
)dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ+ 2aλ
)P2
minusλ2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0xkeminus
2θxλ dx =
λk+1k2k+1θk+1 P2 =
intinfin0xk+1eminus
2θxλ dx =
λk+2(k+ 1)2k+2θk+2
Adetunji AA Transmuted Sushila Distribution 8
P3 =
intinfin0xkeminus
θxλ dx =
λk+1kθk+1 P4 =
intinfin0xk+2eminus
2θxλ dx =
λk+3k2k+3θk+2
P5 =
intinfin0xk+1eminus
θxλ dx =
λk+2(k+ 1)θk+2
Therefore
E(xk) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)(λk+1k
2k+1θk+1
)+
(4aθλ+ 2aλ
)(λk+2(k+ 1)
2k+2θk+2
)minusλ2
(aθ+ aminus θminus 1
)(λk+1kθk+1
)+ 2aθ
(λk+3k
2k+3θk+2
)minusλ
(aθ+ aminus θminus 1
)(λk+2(k+ 1)
θk+2
)
Hence
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1 minus(aθ+aminusθminus1)(θ+k+1)+a(k+ 1)(k+ 2)
2k+2
)
Mean and Variance of TSD The mean of a random variable X follows TSD is obtainedby putting k = 1 in (47) above Hence the mean is given as
E(x) =λ
4θ(1 + θ)2
(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8
)Also the variance of X is obtained as var(x) = E(x2) minus (E(x))2 where E(x2) is obtained byequating k = 2 in (47)
E(x2) =λ2
4θ2(1 + θ)2
(6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24
)Hence the variance of X is given by
var(x) =λ2
4θ2(1 + θ)2
[(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8)2
4(1 + θ)2
+ (6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24)]
Adetunji AA Transmuted Sushila Distribution 9
45 Moment Generating FunctionProposition 42 The moment generating function of a random variable X that followsthe TSD is given as
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3
minus(aθ+ aminus θminus 1)(θminus λt+ 1)
(θminus λt)2
)
(48)
Proof
Mx(t) = E(etx) =
intinfin0etxf(x)dx
=
intinfin0etx
θ2
λ3(1 + θ)2 (λ+ x)
(2aeminus
θxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0(λ+ x)eminus
θminusλtλ x
(2aθλeminus
θxλ + 2aθxeminus
θxλ + 2aλeminus
θxλ
minusλ(aθ+ aminus θminus 1))dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ
+2aλ)P2 minus λ
2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0eminus
2θminusλtλ x dx =
λ
2θminus λt P2 =
intinfin0xeminus
2θminusλtλ x dx =
λ2
(2θminus λt)2
P3 =
intinfin0
eminusθminusλtλ x dx =
λ
(θminus λt) P4 =
intinfin0
x2eminus2θminusλtλ x dx =
2λ3
(2θminus λt)3
P5 =
intinfin0
xeminusθminusλtλ x dx =
λ2
(θminus λt)2
Adetunji AA Transmuted Sushila Distribution 10
Therefore
E(etx) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2)
(λ
2θminus λt
)+ (4aθλ+ 2aλ)
(λ
(2θminus λt)
)2
minusλ2(aθ+ aminus θminus 1)(
λ
θminus λt
)+ 4aθ
(2λ
(2θminus λt)
)3
minus λ(aθ+ aminus θminus 1)(
λ
(θminus λt)
=
(θ2
λ3(1 + θ)2
)[2aλ2(1 + θ)
(λ
2θminus λt
)+2aλ(1 + 2θ)
(λ
2θminus λt
)2
+ 4aθ(
λ
2θminus λt
)3
minus(aθ+ aminus θminus 1)λ3(
1 +1
θminus λt
)]=
(θ2
λ3(1 + θ)2
)[2a(1 + θ)
(1
2θminus λt
)+2a(1 + 2θ)
(1
2θminus λt
)2
+ 4aθ(
12θminus λt
)2
minus(aθ+ aminus θminus 1)λ3(θminus λt+ 1θminus λt
)]
Hence
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3 minus
(aθ+ aminus θminus 1)(θminus λt+ 1)(θminus λt)2
)
5 Parameter Estimation
Proposition 51 Given that Xi i = 1 2 n are iid random variables from TSD then thelog-likelihood function of X is defined as
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
(51)
Adetunji AA Transmuted Sushila Distribution 11
Proof The likelihood function of a random variable X that follows TSD is
L =
i=nprodi=1
(θ2
λ3(1 + θ)2
)(λ+ xi)
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλxi
=
(θ2n
λ3n(1 + θ)2n
) i=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
=
(θ
1 + θ
)2n
λminus3ni=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
Hence the log-likelihood function of a random variable X that follows TSD is
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
The MLE of (θ λa) can be obtained by maximizing (51) This gives the set of normalequations below
partlogL
partθ=
2nθ
minus2n
1 + θminus
1λ
nsumi=1
xi
+
nsumi=1
minus 1λ2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(λ+ xi + λ) minus aλ+ λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
partlogL
partλ=
minus3nθ
+
nsumi=1
1λ+ xi
+θ
λ2
nsumi=1
xi
+
nsumi=1
minus 1λ2 2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(1 + θ) minus aθminus a+ θ+ 1
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 6
If r = 1 and r = n the 1st and nth order statistic for X are respectively given in (42)and (43)
f1n(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][
1 minus (1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)+ a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminusr
(42)
fnn(x) =nθ2(λ+ x)
(nminus 1)
[(2aeminusθλx(λθ+ θx+ λ) minus aθλminus aλ+ θλminus λ
)eminus
θλx
λ3(1 + θ)2
][(1 + a)
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)minus a
(1 minus
λ(1 + θ) + θx
λ(1 + θ)eminus
θλx
)2]nminus1
(43)
42 Quantiles FunctionThe quantile function of a random variable X sim TSD(θ λ a) is given as
Q(u) = minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + 2a+ a2 minus 4ua)2a
)+ 1 + θ
] (44)
Therefore the first the second and the third quartiles for the random variable are ob-tained by respectively setting u to 025 050 and 075 These are given by
Q( 14 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 + a)
2a
)+ 1 + θ
]
Q( 12 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2)
2a
)+ 1 + θ
]
Q( 34 )
= minus
(λ
θ
)[W(minus
eminus(1+θ)(1 + θ)(aminus 1 +radic
1 + a2 minus a)
2a
)+ 1 + θ
]
where the Lambert function W is a complex function with multiple values which is definedas the solution for the equation W(u)e
W(u) = u
Equation (44) is obtained using Maple 2016 [15]
Adetunji AA Transmuted Sushila Distribution 7
43 Skewness and KurtosisVariability in a data set can be investigated using skewness and kurtosis and classical
measures can be susceptible to outliers Given the quantile function as in (44) the MoorrsquosKurtosis [16] based on octiles is given as
KM =Q 7
8minusQ 5
8+Q 3
8minusQ 1
8
Q 68minusQ 2
8
(45)
Also the Bowleyrsquos measure of skewness [17] based on quartiles is given as
SKB =Q 3
4minus 2Q 2
4+Q 1
4
Q 34minusQ 1
4
(46)
44 MomentsProposition 41 If a random variable X follows TSD with pdf as given in (22) then thekth raw moment is given by
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1
minus (aθ+ aminus θminus 1)(θ+ k+ 1) +a(k+ 1)(k+ 2)
2k+2
) (47)
Proof
micro1k = E(xk) =
intinfin0xkf(x)dx
=
intinfin0xk
θ2
λ3(1 + θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(λθ2aeminus
θxλ
+θx2aeminusθxλ + λ2aeminus
θxλ minus aθλminus aλ+ θλ+ λ
)dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ+ 2aλ
)P2
minusλ2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0xkeminus
2θxλ dx =
λk+1k2k+1θk+1 P2 =
intinfin0xk+1eminus
2θxλ dx =
λk+2(k+ 1)2k+2θk+2
Adetunji AA Transmuted Sushila Distribution 8
P3 =
intinfin0xkeminus
θxλ dx =
λk+1kθk+1 P4 =
intinfin0xk+2eminus
2θxλ dx =
λk+3k2k+3θk+2
P5 =
intinfin0xk+1eminus
θxλ dx =
λk+2(k+ 1)θk+2
Therefore
E(xk) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)(λk+1k
2k+1θk+1
)+
(4aθλ+ 2aλ
)(λk+2(k+ 1)
2k+2θk+2
)minusλ2
(aθ+ aminus θminus 1
)(λk+1kθk+1
)+ 2aθ
(λk+3k
2k+3θk+2
)minusλ
(aθ+ aminus θminus 1
)(λk+2(k+ 1)
θk+2
)
Hence
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1 minus(aθ+aminusθminus1)(θ+k+1)+a(k+ 1)(k+ 2)
2k+2
)
Mean and Variance of TSD The mean of a random variable X follows TSD is obtainedby putting k = 1 in (47) above Hence the mean is given as
E(x) =λ
4θ(1 + θ)2
(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8
)Also the variance of X is obtained as var(x) = E(x2) minus (E(x))2 where E(x2) is obtained byequating k = 2 in (47)
E(x2) =λ2
4θ2(1 + θ)2
(6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24
)Hence the variance of X is given by
var(x) =λ2
4θ2(1 + θ)2
[(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8)2
4(1 + θ)2
+ (6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24)]
Adetunji AA Transmuted Sushila Distribution 9
45 Moment Generating FunctionProposition 42 The moment generating function of a random variable X that followsthe TSD is given as
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3
minus(aθ+ aminus θminus 1)(θminus λt+ 1)
(θminus λt)2
)
(48)
Proof
Mx(t) = E(etx) =
intinfin0etxf(x)dx
=
intinfin0etx
θ2
λ3(1 + θ)2 (λ+ x)
(2aeminus
θxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0(λ+ x)eminus
θminusλtλ x
(2aθλeminus
θxλ + 2aθxeminus
θxλ + 2aλeminus
θxλ
minusλ(aθ+ aminus θminus 1))dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ
+2aλ)P2 minus λ
2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0eminus
2θminusλtλ x dx =
λ
2θminus λt P2 =
intinfin0xeminus
2θminusλtλ x dx =
λ2
(2θminus λt)2
P3 =
intinfin0
eminusθminusλtλ x dx =
λ
(θminus λt) P4 =
intinfin0
x2eminus2θminusλtλ x dx =
2λ3
(2θminus λt)3
P5 =
intinfin0
xeminusθminusλtλ x dx =
λ2
(θminus λt)2
Adetunji AA Transmuted Sushila Distribution 10
Therefore
E(etx) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2)
(λ
2θminus λt
)+ (4aθλ+ 2aλ)
(λ
(2θminus λt)
)2
minusλ2(aθ+ aminus θminus 1)(
λ
θminus λt
)+ 4aθ
(2λ
(2θminus λt)
)3
minus λ(aθ+ aminus θminus 1)(
λ
(θminus λt)
=
(θ2
λ3(1 + θ)2
)[2aλ2(1 + θ)
(λ
2θminus λt
)+2aλ(1 + 2θ)
(λ
2θminus λt
)2
+ 4aθ(
λ
2θminus λt
)3
minus(aθ+ aminus θminus 1)λ3(
1 +1
θminus λt
)]=
(θ2
λ3(1 + θ)2
)[2a(1 + θ)
(1
2θminus λt
)+2a(1 + 2θ)
(1
2θminus λt
)2
+ 4aθ(
12θminus λt
)2
minus(aθ+ aminus θminus 1)λ3(θminus λt+ 1θminus λt
)]
Hence
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3 minus
(aθ+ aminus θminus 1)(θminus λt+ 1)(θminus λt)2
)
5 Parameter Estimation
Proposition 51 Given that Xi i = 1 2 n are iid random variables from TSD then thelog-likelihood function of X is defined as
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
(51)
Adetunji AA Transmuted Sushila Distribution 11
Proof The likelihood function of a random variable X that follows TSD is
L =
i=nprodi=1
(θ2
λ3(1 + θ)2
)(λ+ xi)
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλxi
=
(θ2n
λ3n(1 + θ)2n
) i=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
=
(θ
1 + θ
)2n
λminus3ni=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
Hence the log-likelihood function of a random variable X that follows TSD is
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
The MLE of (θ λa) can be obtained by maximizing (51) This gives the set of normalequations below
partlogL
partθ=
2nθ
minus2n
1 + θminus
1λ
nsumi=1
xi
+
nsumi=1
minus 1λ2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(λ+ xi + λ) minus aλ+ λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
partlogL
partλ=
minus3nθ
+
nsumi=1
1λ+ xi
+θ
λ2
nsumi=1
xi
+
nsumi=1
minus 1λ2 2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(1 + θ) minus aθminus a+ θ+ 1
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 7
43 Skewness and KurtosisVariability in a data set can be investigated using skewness and kurtosis and classical
measures can be susceptible to outliers Given the quantile function as in (44) the MoorrsquosKurtosis [16] based on octiles is given as
KM =Q 7
8minusQ 5
8+Q 3
8minusQ 1
8
Q 68minusQ 2
8
(45)
Also the Bowleyrsquos measure of skewness [17] based on quartiles is given as
SKB =Q 3
4minus 2Q 2
4+Q 1
4
Q 34minusQ 1
4
(46)
44 MomentsProposition 41 If a random variable X follows TSD with pdf as given in (22) then thekth raw moment is given by
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1
minus (aθ+ aminus θminus 1)(θ+ k+ 1) +a(k+ 1)(k+ 2)
2k+2
) (47)
Proof
micro1k = E(xk) =
intinfin0xkf(x)dx
=
intinfin0xk
θ2
λ3(1 + θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)dx
=
(θ2
λ3(1 + θ)2
) intinfin0
(λxkeminus
θxλ + xk+1eminus
θxλ
)(λθ2aeminus
θxλ
+θx2aeminusθxλ + λ2aeminus
θxλ minus aθλminus aλ+ θλ+ λ
)dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ+ 2aλ
)P2
minusλ2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0xkeminus
2θxλ dx =
λk+1k2k+1θk+1 P2 =
intinfin0xk+1eminus
2θxλ dx =
λk+2(k+ 1)2k+2θk+2
Adetunji AA Transmuted Sushila Distribution 8
P3 =
intinfin0xkeminus
θxλ dx =
λk+1kθk+1 P4 =
intinfin0xk+2eminus
2θxλ dx =
λk+3k2k+3θk+2
P5 =
intinfin0xk+1eminus
θxλ dx =
λk+2(k+ 1)θk+2
Therefore
E(xk) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)(λk+1k
2k+1θk+1
)+
(4aθλ+ 2aλ
)(λk+2(k+ 1)
2k+2θk+2
)minusλ2
(aθ+ aminus θminus 1
)(λk+1kθk+1
)+ 2aθ
(λk+3k
2k+3θk+2
)minusλ
(aθ+ aminus θminus 1
)(λk+2(k+ 1)
θk+2
)
Hence
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1 minus(aθ+aminusθminus1)(θ+k+1)+a(k+ 1)(k+ 2)
2k+2
)
Mean and Variance of TSD The mean of a random variable X follows TSD is obtainedby putting k = 1 in (47) above Hence the mean is given as
E(x) =λ
4θ(1 + θ)2
(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8
)Also the variance of X is obtained as var(x) = E(x2) minus (E(x))2 where E(x2) is obtained byequating k = 2 in (47)
E(x2) =λ2
4θ2(1 + θ)2
(6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24
)Hence the variance of X is given by
var(x) =λ2
4θ2(1 + θ)2
[(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8)2
4(1 + θ)2
+ (6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24)]
Adetunji AA Transmuted Sushila Distribution 9
45 Moment Generating FunctionProposition 42 The moment generating function of a random variable X that followsthe TSD is given as
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3
minus(aθ+ aminus θminus 1)(θminus λt+ 1)
(θminus λt)2
)
(48)
Proof
Mx(t) = E(etx) =
intinfin0etxf(x)dx
=
intinfin0etx
θ2
λ3(1 + θ)2 (λ+ x)
(2aeminus
θxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0(λ+ x)eminus
θminusλtλ x
(2aθλeminus
θxλ + 2aθxeminus
θxλ + 2aλeminus
θxλ
minusλ(aθ+ aminus θminus 1))dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ
+2aλ)P2 minus λ
2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0eminus
2θminusλtλ x dx =
λ
2θminus λt P2 =
intinfin0xeminus
2θminusλtλ x dx =
λ2
(2θminus λt)2
P3 =
intinfin0
eminusθminusλtλ x dx =
λ
(θminus λt) P4 =
intinfin0
x2eminus2θminusλtλ x dx =
2λ3
(2θminus λt)3
P5 =
intinfin0
xeminusθminusλtλ x dx =
λ2
(θminus λt)2
Adetunji AA Transmuted Sushila Distribution 10
Therefore
E(etx) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2)
(λ
2θminus λt
)+ (4aθλ+ 2aλ)
(λ
(2θminus λt)
)2
minusλ2(aθ+ aminus θminus 1)(
λ
θminus λt
)+ 4aθ
(2λ
(2θminus λt)
)3
minus λ(aθ+ aminus θminus 1)(
λ
(θminus λt)
=
(θ2
λ3(1 + θ)2
)[2aλ2(1 + θ)
(λ
2θminus λt
)+2aλ(1 + 2θ)
(λ
2θminus λt
)2
+ 4aθ(
λ
2θminus λt
)3
minus(aθ+ aminus θminus 1)λ3(
1 +1
θminus λt
)]=
(θ2
λ3(1 + θ)2
)[2a(1 + θ)
(1
2θminus λt
)+2a(1 + 2θ)
(1
2θminus λt
)2
+ 4aθ(
12θminus λt
)2
minus(aθ+ aminus θminus 1)λ3(θminus λt+ 1θminus λt
)]
Hence
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3 minus
(aθ+ aminus θminus 1)(θminus λt+ 1)(θminus λt)2
)
5 Parameter Estimation
Proposition 51 Given that Xi i = 1 2 n are iid random variables from TSD then thelog-likelihood function of X is defined as
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
(51)
Adetunji AA Transmuted Sushila Distribution 11
Proof The likelihood function of a random variable X that follows TSD is
L =
i=nprodi=1
(θ2
λ3(1 + θ)2
)(λ+ xi)
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλxi
=
(θ2n
λ3n(1 + θ)2n
) i=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
=
(θ
1 + θ
)2n
λminus3ni=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
Hence the log-likelihood function of a random variable X that follows TSD is
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
The MLE of (θ λa) can be obtained by maximizing (51) This gives the set of normalequations below
partlogL
partθ=
2nθ
minus2n
1 + θminus
1λ
nsumi=1
xi
+
nsumi=1
minus 1λ2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(λ+ xi + λ) minus aλ+ λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
partlogL
partλ=
minus3nθ
+
nsumi=1
1λ+ xi
+θ
λ2
nsumi=1
xi
+
nsumi=1
minus 1λ2 2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(1 + θ) minus aθminus a+ θ+ 1
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 8
P3 =
intinfin0xkeminus
θxλ dx =
λk+1kθk+1 P4 =
intinfin0xk+2eminus
2θxλ dx =
λk+3k2k+3θk+2
P5 =
intinfin0xk+1eminus
θxλ dx =
λk+2(k+ 1)θk+2
Therefore
E(xk) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)(λk+1k
2k+1θk+1
)+
(4aθλ+ 2aλ
)(λk+2(k+ 1)
2k+2θk+2
)minusλ2
(aθ+ aminus θminus 1
)(λk+1kθk+1
)+ 2aθ
(λk+3k
2k+3θk+2
)minusλ
(aθ+ aminus θminus 1
)(λk+2(k+ 1)
θk+2
)
Hence
E(xk) =λkk
θk(1 + θ)2
(aθ(1 + θ)
2k+a(1 + 2θ)(k+ 1)
2k+1 minus(aθ+aminusθminus1)(θ+k+1)+a(k+ 1)(k+ 2)
2k+2
)
Mean and Variance of TSD The mean of a random variable X follows TSD is obtainedby putting k = 1 in (47) above Hence the mean is given as
E(x) =λ
4θ(1 + θ)2
(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8
)Also the variance of X is obtained as var(x) = E(x2) minus (E(x))2 where E(x2) is obtained byequating k = 2 in (47)
E(x2) =λ2
4θ2(1 + θ)2
(6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24
)Hence the variance of X is given by
var(x) =λ2
4θ2(1 + θ)2
[(2aθ2 + 6aθminus 4θ2 + 3aminus 12θminus 8)2
4(1 + θ)2
+ (6aθ2 + 24aθminus 8θ2 + 15aminus 32θminus 24)]
Adetunji AA Transmuted Sushila Distribution 9
45 Moment Generating FunctionProposition 42 The moment generating function of a random variable X that followsthe TSD is given as
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3
minus(aθ+ aminus θminus 1)(θminus λt+ 1)
(θminus λt)2
)
(48)
Proof
Mx(t) = E(etx) =
intinfin0etxf(x)dx
=
intinfin0etx
θ2
λ3(1 + θ)2 (λ+ x)
(2aeminus
θxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0(λ+ x)eminus
θminusλtλ x
(2aθλeminus
θxλ + 2aθxeminus
θxλ + 2aλeminus
θxλ
minusλ(aθ+ aminus θminus 1))dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ
+2aλ)P2 minus λ
2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0eminus
2θminusλtλ x dx =
λ
2θminus λt P2 =
intinfin0xeminus
2θminusλtλ x dx =
λ2
(2θminus λt)2
P3 =
intinfin0
eminusθminusλtλ x dx =
λ
(θminus λt) P4 =
intinfin0
x2eminus2θminusλtλ x dx =
2λ3
(2θminus λt)3
P5 =
intinfin0
xeminusθminusλtλ x dx =
λ2
(θminus λt)2
Adetunji AA Transmuted Sushila Distribution 10
Therefore
E(etx) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2)
(λ
2θminus λt
)+ (4aθλ+ 2aλ)
(λ
(2θminus λt)
)2
minusλ2(aθ+ aminus θminus 1)(
λ
θminus λt
)+ 4aθ
(2λ
(2θminus λt)
)3
minus λ(aθ+ aminus θminus 1)(
λ
(θminus λt)
=
(θ2
λ3(1 + θ)2
)[2aλ2(1 + θ)
(λ
2θminus λt
)+2aλ(1 + 2θ)
(λ
2θminus λt
)2
+ 4aθ(
λ
2θminus λt
)3
minus(aθ+ aminus θminus 1)λ3(
1 +1
θminus λt
)]=
(θ2
λ3(1 + θ)2
)[2a(1 + θ)
(1
2θminus λt
)+2a(1 + 2θ)
(1
2θminus λt
)2
+ 4aθ(
12θminus λt
)2
minus(aθ+ aminus θminus 1)λ3(θminus λt+ 1θminus λt
)]
Hence
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3 minus
(aθ+ aminus θminus 1)(θminus λt+ 1)(θminus λt)2
)
5 Parameter Estimation
Proposition 51 Given that Xi i = 1 2 n are iid random variables from TSD then thelog-likelihood function of X is defined as
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
(51)
Adetunji AA Transmuted Sushila Distribution 11
Proof The likelihood function of a random variable X that follows TSD is
L =
i=nprodi=1
(θ2
λ3(1 + θ)2
)(λ+ xi)
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλxi
=
(θ2n
λ3n(1 + θ)2n
) i=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
=
(θ
1 + θ
)2n
λminus3ni=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
Hence the log-likelihood function of a random variable X that follows TSD is
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
The MLE of (θ λa) can be obtained by maximizing (51) This gives the set of normalequations below
partlogL
partθ=
2nθ
minus2n
1 + θminus
1λ
nsumi=1
xi
+
nsumi=1
minus 1λ2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(λ+ xi + λ) minus aλ+ λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
partlogL
partλ=
minus3nθ
+
nsumi=1
1λ+ xi
+θ
λ2
nsumi=1
xi
+
nsumi=1
minus 1λ2 2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(1 + θ) minus aθminus a+ θ+ 1
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 9
45 Moment Generating FunctionProposition 42 The moment generating function of a random variable X that followsthe TSD is given as
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3
minus(aθ+ aminus θminus 1)(θminus λt+ 1)
(θminus λt)2
)
(48)
Proof
Mx(t) = E(etx) =
intinfin0etxf(x)dx
=
intinfin0etx
θ2
λ3(1 + θ)2 (λ+ x)
(2aeminus
θxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ dx
=
(θ2
λ3(1 + θ)2
) intinfin0(λ+ x)eminus
θminusλtλ x
(2aθλeminus
θxλ + 2aθxeminus
θxλ + 2aλeminus
θxλ
minusλ(aθ+ aminus θminus 1))dx
=
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2
)P1 +
(4aθλ
+2aλ)P2 minus λ
2(aθ+ aminus θminus 1
)P3 + 2aθP4 minus λ
(aθ+ aminus θminus 1
)P5
where
P1 =
intinfin0eminus
2θminusλtλ x dx =
λ
2θminus λt P2 =
intinfin0xeminus
2θminusλtλ x dx =
λ2
(2θminus λt)2
P3 =
intinfin0
eminusθminusλtλ x dx =
λ
(θminus λt) P4 =
intinfin0
x2eminus2θminusλtλ x dx =
2λ3
(2θminus λt)3
P5 =
intinfin0
xeminusθminusλtλ x dx =
λ2
(θminus λt)2
Adetunji AA Transmuted Sushila Distribution 10
Therefore
E(etx) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2)
(λ
2θminus λt
)+ (4aθλ+ 2aλ)
(λ
(2θminus λt)
)2
minusλ2(aθ+ aminus θminus 1)(
λ
θminus λt
)+ 4aθ
(2λ
(2θminus λt)
)3
minus λ(aθ+ aminus θminus 1)(
λ
(θminus λt)
=
(θ2
λ3(1 + θ)2
)[2aλ2(1 + θ)
(λ
2θminus λt
)+2aλ(1 + 2θ)
(λ
2θminus λt
)2
+ 4aθ(
λ
2θminus λt
)3
minus(aθ+ aminus θminus 1)λ3(
1 +1
θminus λt
)]=
(θ2
λ3(1 + θ)2
)[2a(1 + θ)
(1
2θminus λt
)+2a(1 + 2θ)
(1
2θminus λt
)2
+ 4aθ(
12θminus λt
)2
minus(aθ+ aminus θminus 1)λ3(θminus λt+ 1θminus λt
)]
Hence
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3 minus
(aθ+ aminus θminus 1)(θminus λt+ 1)(θminus λt)2
)
5 Parameter Estimation
Proposition 51 Given that Xi i = 1 2 n are iid random variables from TSD then thelog-likelihood function of X is defined as
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
(51)
Adetunji AA Transmuted Sushila Distribution 11
Proof The likelihood function of a random variable X that follows TSD is
L =
i=nprodi=1
(θ2
λ3(1 + θ)2
)(λ+ xi)
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλxi
=
(θ2n
λ3n(1 + θ)2n
) i=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
=
(θ
1 + θ
)2n
λminus3ni=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
Hence the log-likelihood function of a random variable X that follows TSD is
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
The MLE of (θ λa) can be obtained by maximizing (51) This gives the set of normalequations below
partlogL
partθ=
2nθ
minus2n
1 + θminus
1λ
nsumi=1
xi
+
nsumi=1
minus 1λ2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(λ+ xi + λ) minus aλ+ λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
partlogL
partλ=
minus3nθ
+
nsumi=1
1λ+ xi
+θ
λ2
nsumi=1
xi
+
nsumi=1
minus 1λ2 2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(1 + θ) minus aθminus a+ θ+ 1
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 10
Therefore
E(etx) =
(θ2
λ3(1 + θ)2
)(2aθλ2 + 2aλ2)
(λ
2θminus λt
)+ (4aθλ+ 2aλ)
(λ
(2θminus λt)
)2
minusλ2(aθ+ aminus θminus 1)(
λ
θminus λt
)+ 4aθ
(2λ
(2θminus λt)
)3
minus λ(aθ+ aminus θminus 1)(
λ
(θminus λt)
=
(θ2
λ3(1 + θ)2
)[2aλ2(1 + θ)
(λ
2θminus λt
)+2aλ(1 + 2θ)
(λ
2θminus λt
)2
+ 4aθ(
λ
2θminus λt
)3
minus(aθ+ aminus θminus 1)λ3(
1 +1
θminus λt
)]=
(θ2
λ3(1 + θ)2
)[2a(1 + θ)
(1
2θminus λt
)+2a(1 + 2θ)
(1
2θminus λt
)2
+ 4aθ(
12θminus λt
)2
minus(aθ+ aminus θminus 1)λ3(θminus λt+ 1θminus λt
)]
Hence
E(etx) =
(θ2
(1 + θ)2
)(2a(1 + θ)
2θminus λt+
2a(1 + 2θ)(2θminus λt)2 +
4aθ(2θminus λt)3 minus
(aθ+ aminus θminus 1)(θminus λt+ 1)(θminus λt)2
)
5 Parameter Estimation
Proposition 51 Given that Xi i = 1 2 n are iid random variables from TSD then thelog-likelihood function of X is defined as
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
(51)
Adetunji AA Transmuted Sushila Distribution 11
Proof The likelihood function of a random variable X that follows TSD is
L =
i=nprodi=1
(θ2
λ3(1 + θ)2
)(λ+ xi)
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλxi
=
(θ2n
λ3n(1 + θ)2n
) i=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
=
(θ
1 + θ
)2n
λminus3ni=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
Hence the log-likelihood function of a random variable X that follows TSD is
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
The MLE of (θ λa) can be obtained by maximizing (51) This gives the set of normalequations below
partlogL
partθ=
2nθ
minus2n
1 + θminus
1λ
nsumi=1
xi
+
nsumi=1
minus 1λ2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(λ+ xi + λ) minus aλ+ λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
partlogL
partλ=
minus3nθ
+
nsumi=1
1λ+ xi
+θ
λ2
nsumi=1
xi
+
nsumi=1
minus 1λ2 2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(1 + θ) minus aθminus a+ θ+ 1
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 11
Proof The likelihood function of a random variable X that follows TSD is
L =
i=nprodi=1
(θ2
λ3(1 + θ)2
)(λ+ xi)
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλxi
=
(θ2n
λ3n(1 + θ)2n
) i=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
=
(θ
1 + θ
)2n
λminus3ni=nprodi=1
(λ+ xi)
i=nprodi=1
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλminus aθλminus aλ+ θλ+ λ
)eminus
θλ
sumni=1 xi
Hence the log-likelihood function of a random variable X that follows TSD is
logL = 2n(log(θ) minus log(1 + θ)
)minus 3nlog(λ) +
nsumi=1
log(λ+ xi)
+
nsumi=1
log
(2aeminus
θλxi(λθ+ θxi + λ) minus aθλ+ θλ+ λ
)minusλ
θ
nsumi=1
xi
The MLE of (θ λa) can be obtained by maximizing (51) This gives the set of normalequations below
partlogL
partθ=
2nθ
minus2n
1 + θminus
1λ
nsumi=1
xi
+
nsumi=1
minus 1λ2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(λ+ xi + λ) minus aλ+ λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
partlogL
partλ=
minus3nθ
+
nsumi=1
1λ+ xi
+θ
λ2
nsumi=1
xi
+
nsumi=1
minus 1λ2 2aeminus
θλxi(λθ+ θxi + λ) + 2aeminus
θλxi(1 + θ) minus aθminus a+ θ+ 1
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 12
partlogL
parta=
nsumi=1
2aeminusθλxi(λθ+ θxi + λ) minus θλminus λ
2aeminusθλxi(λθ+ θxi + λ) minus aθλminus aλ+ θλ+ λ
= 0
Obtaining solutions for the set of normal equations analytically is tedious Using theMaxLik function in R language [18] the solutions are obtained numerically using algo-rithms like Newton-Raphson
51 Asymptotic Confidence Bounds of TSDUsing the variance-covariance matrix Iminus1
o a (100 minus α) confidence intervals of theparameters θ λa can be obtained Iminus1
n [Ψ] is the inverse of the observed informationmatrix [19] given by
Iminus1n [Ψ] =
[ Iθθ Iθλ IθaIλθ Iλλ IλaIaθ Iaλ Iaa
]=
[ var(θ) covar(θ λ) covar(θ a)covar(θ λ) var(λ) covar(λ a)covar(θ a) covar(λ a) var(a)
]
Iθθ = part2logLpartθ2 Iλλ = part2logL
partλ2 Iaa = part2logLparta2 Iθλ = part2logL
partθpartλ Iθa = part2logLpartθparta Iaλ = part2logL
partapartλ
Therefore at specified level of significance (100minusα) confidence intervals of (θ λa)
are respectively given as θplusmnZα2
radicvar(θ) λplusmnZα
2
radicvar(λ) aplusmnZα
2
radicvar(a)
6 Application
The QT technique has been applied to obtain new set of distributions by compound-ing the Lindley distribution The Transmuted Lindley distribution (TLD) was introducedby [20] [21] introduced Transmuted Quasi-Lindley distribution (TQLD) while [22] in-troduced the Transmuted Two-Parameter Lindley distribution (TTPLD) The TransmutedGeneralized Quasi Lindley distribution (TGQLD) was introduced by [23] All these newlytransmuted distributions have the Lindley distribution has special case This attribute isalso shared by the TSD newly introduced in this research The probability distributionfunctions of all the models compared for data application are presented in table 1 below
The data set used to observe the performance of TSD is the remission times (months)of a sample of 128 bladder cancer patients This data has been applied in various survivalanalysis [19 20]
008 209 348487 694 866 1311 2363 020 223 352 498 697 902 1329 040 226 357506 709
922 1380 2574 050 246 364 509 726 947 1424 2582 051 254370 517 728 974 1476 2631 081
262 382 532 732 1006 1477 3215 264388 532 739 1034 1483 3426 090 269 418 534 759 1066
1596 3666 105269 423 541 762 1075 1662 4301 119 275 426 541 763 1712 4612 126283 433
549 766 1125 1714 7905 135 287 562 787 1164 1736 140 302434 571 793 1179 1810 146 440
585 826 1198 1913 176 325 450 625837 1202 202 331 451 654 853 1203 2028 202 336 676
1207 2173 207336 693 865 1263 2269
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 13
Table 1 Probability distribution functions of compared modelsDistribution PDF
TSD θ2
λ3(1+θ)2 (λ+ x)(
2aeminusθxλ (λθ+ θx+ λ) minus λ(1 + θ)(aminus 1)
)eminus
θxλ
SD θ2
λ(1+θ)
(1 + x
λ
)eminus
θλ
x
LD θ2
1+θ (1 + x)eminusθλ
x
TLD θ2
1+θ (1 + x)eminusθλ
x(
1 minus λ+ 2λ 1+θ+θx1+θ
)TQLD θ
1+α (α+ θx)eminusθx(
1 minus λ+ 2λeminusθx(θx
1+α
))TTPLD θ2
θ+α (1 +αx)eminusθx(
1 + λminus 2λ(
1 minus θ+α+αθxθ+α eminusθx
))TGQLD aθ
1+α (α+ θx)eminusθx(
1 minus eminusθx(
1 + θx1+α
))aminus1(1 + λminus 2λ
(1 minus θx
1+α eminusθx)α)
Table 2 Descriptive Statistics for remission times in monthsMin Q1 Median Q3 Max Mean Variance Skewness Kurtosis0080 3348 6395 11838 79050 9366 110425 3287 15483
The descriptive statistics for the data set is presented in table 2 while table 3 showsthe parameter estimates and model comparison criteria To compare the models -2LL(negative 2 Log-Likelihood) AIC (Akaike Information Criterion) and CAIC (CorrectedAkaike Information Criterion) are used Distribution with lowest criterion is the bestResults from table 3 shows that the newly proposed distribution (TSD) performs creditablywell for the lifetime dataset presented
7 Conclusion
We introduced a three-parameter generalization of the Sushila distribution using theQuadratic Transmuted technique championed by [3] Shape of the distribution functionand hazard rate function are investigated and various mathematical properties of the newdistribution are presented Since the Sushila distribution [9] is a special case of the Lindleydistribution performance of the new distribution is compared with the Sushila distributionand other transmuted Lindley distributions using data on cancer remission (in moths) ofpatients Results show that the Transmuted Sushila Distribution (TSD) givers a better fitto the data set among the competing distributions
Competing Interest
Author has declared that no competing interest exist
References
[1] Marshall AW Olkin IA (1997) New Method for Adding a Parameter to a Family of Distri-butions with Application to the Exponential and Weibull Families Biometrika 84 641-652httpsdoi101093biomet843641
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-
Adetunji AA Transmuted Sushila Distribution 14
Table 3 Parameter estimates and model selection criteriaθ λ a α -2 LL AIC CAIC
TSD 0742 6323 0712 825884 831884 832037SD 52650 484078 828694 832694 832766LD 0196 839040 841060 841064TLD 0156 0617 830310 834310 834382TQLD 0061 1138000 0862 826994 832995 833147TTPLD 0152 0531 828087 832087 832159TGQLD 0121 1218 0500 1534000 826149 834149 834417
[2] Eugene N Lee C Famoye F (2002) Beta-Normal Distribution and Its Applications Communications inStatistics-Theory and Methods 31497-512 httpsdoi101081sta-120003130
[3] Shaw W Buckley I (2007) The alchemy of probability distributions beyond Gram-Charlier expansionsand a skew-kurtotic-normal distribution from a rank transmutation map Research report
[4] Cordeiro GM Castro M (2011) A New Family of Generalized Distributions Journal of Statistical Compu-tation and Simulation 81883-898 httpsdoi10108000949650903530745
[5] Ristic MM Balakrishnan N (2012) The Gamma Exponentiated Exponential Distribution Journal of Sta-tistical Computation and Simulation 821191-1206 httpsdoi101080009496552011574633
[6] Cordeiro GM Ortega EMM Cunha DCC (2013) The Exponentiated Generalized Class of DistributionsJournal of Data Science 111-27
[7] Alzaatreh A Lee C Famoye F (2013) A New Method for Generating Families of Continuous DistributionsMetron 7163-79 httpsdoi101007s40300-013-0007-y
[8] Mahdavi A Kundu D (2017) A new method for generating distributions with an application to exponentialdistribution Communications in Statistics-Theory and Methods 46(13)6543-6557
[9] Shanker R Shambhu S Uma S Ravi S (2013) Sushila distribution and its application to waiting timesdata International Journal of Business Management 3(2)1 -11
[10] Lindley DV (1958) Fiducial distributions and Bayes-Theorem Journal of the Royal Statistical SocietyB(20)102-107
[11] Aryal GR Tsokos CP (2009) On the transmuted extreme value distribution with application NonlinearAnalysis Theory Methods and Applications 711401-1407 httpsdoi101016jna200901168
[12] Aryal GR Tsokos CP (2011) Transmuted Weibull distribution A generalization of the Weibull probabilitydistribution European Journal of Pure and Applied Mathematics 489-102
[13] Bourguignon M Ghosh I Cordeiro GM (2016) General Results for the Transmuted Family of Distributionsand New Models Journal of Probability and Statistics 7208425 httpsdoi10115520167208425
[14] Tahir MH Cordeiro GM (2016) Compounding of distributions a survey and new generalized classes Jour-nal of Statistical Distributions and Applications 31-35 httpsdoiorg101186s40488-016-0052-1
[15] Maple Incorporated a division of Waterloo Maple Incorporated 2016[16] Moors JJA (1998) A quantile alternative for kurtosis The Statistician 3725-32[17] Kenney J Keeping E (1962) Mathematics of Statistics Volume 1 Third edition Princeton Van Nostrand
New Jersey[18] R Development Core Team (2009) R A Language and Environment for Statistical Computing Foundation
for Statistical Computing Vienna Austria[19] Lawless JF (2003) Statistical Models and Methods for lifetime data John Wiley and Sons New York
201108-1113[20] Merovci F (2013) Transmuted Lindley distribution Int J Open Problems Comput Math 663-72[21] Elbatal I Elgarhy M (2013) Transmuted quasi-Lindley distribution A Generalization of the Quasi Lindley
distribution Int J Pure Appl Sci Technol 1859-70[22] Al-khazaleh M Al-Omari AI Al-khazaleh AMH (2016) Transmuted Two-Parameter
Lindley Distribution Journal of Statistics Applications amp Probability 5(3)421-432httpdxdoiorg1018576jsap050306
[23] Elgarhy M Elbatal I Diab LS Hwas HK Shawki AW (2017) Transmuted Generalized Quasi LindleyDistribution International Journal of Scientific Engineering and Science 1(7)1-8
- 1 Introduction
- 2 Transmuted Sushila Distribution
- 3 Reliability Analysis
- 4 Mathematical Properties
-
- 41 Order Statistics
- 42 Quantiles Function
- 43 Skewness and Kurtosis
- 44 Moments
- 45 Moment Generating Function
-
- 5 Parameter Estimation
-
- 51 Asymptotic Confidence Bounds of TSD
-
- 6 Application
- 7 Conclusion
-