parameterestimationbasedonthefrechetprogressive...

Hindawi Publishing CorporationInternational Journal of Quality, Statistics, and ReliabilityVolume 2012, Article ID 245910, 5 pagesdoi:10.1155/2012/245910

Research Article

Parameter Estimation Based on the Frechet ProgressiveType II Censored Data with Binomial Removals

Mohamed Mubarak1, 2

1 Mathematics Department, Faculty of Science, Minia University, El-Minia 61519, Egypt2 Mathematics Department, University College in Lieth, Umm Al-Qura University, Makkah 311, Saudi Arabia

Correspondence should be addressed to Mohamed Mubarak, mubarak [email protected]

Received 17 March 2011; Revised 16 June 2011; Accepted 17 June 2011

Academic Editor: Chun-Ping Lin

Copyright © 2012 Mohamed Mubarak. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper considers the estimation problem for the Frechet distribution under progressive Type II censoring with randomremovals, where the number of units removed at each failure time has a binomial distribution. We use the maximum likelihoodmethod to obtain the estimators of parameters and derive the sampling distributions of the estimators, and we also construct theconfidence intervals for the parameters and percentile of the failure time distribution.

1. Introduction

Recently, the extreme value distribution is becoming increas-ingly important in engineering statistics as a suitable modelto represent phenomena with usually large maximum obser-vations. In engineering circles, this distribution is often calledthe Frechet model. It is one of the pioneers of extreme valuestatistics. The Frechet (extreme value type II) distributionis one of the probability distributions used to modelextreme events. The generalization of the standard Frechetdistribution has been introduced by Nadarajah and Kotz[1] and Abd-Elfattah and omima [2]. There are over fiftyapplications ranging from accelerated life testing through toearthquakes, floods, rain fall, queues in supermarkets, seacurrents, wind speeds, and track race records, see Kotz andNadarajah [3]. Censoring arises in a life test when exactlifetimes are known for only a portion of test units and theremainder of the lifetimes are known only to exceed certainvalues under an experiment. There are several types ofcensored test. One of the most common censoring schemesis Type II censoring. In a Type II censoring, a total of nunits is placed on test, but instead of continuing until alln units have failed, the test is terminated at the time ofthe mth (1 ≤ m ≤ n) unit failure. Type II censoring

with different failure time distribution has been studied bymany authors including Mann et al. [4], Lawless [5], andMeeker and Escobar [6]. If an experiment desires to removelive units at points other than the final termination pointof the life test, the above described scheme will not beof use to experimenter. Type II censoring does not allowfor units to be lost or removed from the test at pointsother than the final termination point see Balakrishnan andAggarwala [7, Chapter 1].

A generalization of Type II censoring is progressive TypeII censoring. Under this scheme, n units are placed on test attime zero, and m failure are going to be observed. When thefirst failure is observed, r1 of surviving units are randomlyselected, removed, and so on. This experiment terminates atthe time when the mth failures is observed and remainingrm = n − r1 − r2 − · · · − rm−1 − m surviving units areall removed. The statistical inference on the parameters offailure time distribution under progressive Type II censoringhas been studied by several authors [7–10]. Note that, inthis scheme, r1, . . . , rm are all prefixed. However, in somepractical situations, these numbers may occur at random[11]. In some reliability experiments, an experimenter maydecide that it is inappropriate or too dangerous to carryon the testing on some of the tested units even though

2 International Journal of Quality, Statistics, and Reliability

these units have not failed. In these cases, the pattern ofremoval at each failure is random. We assume that, any testunit being dropped out from the life test is independent ofthe others but with the same probability p. Then Tea et al.[12] indicated that the number of test units removed at eachfailure time has a binomial distribution.

In this paper, we will make inference on the param-eters of three-parameter Frechet distribution under pro-gressive type II censoring with binomial removals. Themaximum likelihood estimators (MLEs) for the param-eters in an explicit and implicit form are obtained inSection 3. Section 4 discusses the sampling distribution ofthe MLEs and constructs the confidence intervals for theparameters. The estimator and confidence interval for thepercentile of failure time distribution are also presented inSection 4.

2. Description of the Model

Let random variable X has a three-parameter Frechet dis-tribution. The probability density function and cumulativedistribution function are

f (x) =(α

β

)(β

x − θ

)α+1

exp

[−(

β

x − θ

)α],

(x > θ,α,β > 0

),

(1)

F(x) = exp

[−(

β

x − θ

)α],

(x > θ,α,β > 0

), (2)

respectively, where α is continuous shape parameter, β iscontinuous scale parameter, and θ is continuous locationparameter (θ = 0 yields the two-parameter Frechet distri-bution).

3. Estimation of Parameters

Suppose n independent units are placed on a test withthe corresponding life times being identically distributedwith probability density function fX(x) and cumulativedistribution function FX(x). For simplicity of notation,let X1,X2, . . . ,Xm denote a progressively Type II censoredsample. Then, the joint probability density function of all mprogressively Type II-censored order statistics is

fX1,...,Xm(x1, . . . , xm)

= km∏i=1

fX(xi)[1− FX(xi)]ri , x1 < · · · < xm,(3)

where k = n(n − r1 − 1) · · · (n − r1 − · · · − rm−1 −m + 1).Thus, for a progressive Type II with predetermined number

of removals R = r, the conditional likelihood and Log-likelihood functions, respectively, can be written as

L(α,β, θ | R = r

)

= kαm

βm

m∏i=1

[β

xi − θ

](α+1)

×[

1− exp

(−[

β

xi − θ

]α)]ri

exp

(−[

β

xi − θ

]α),

(4)

L∗(α,β, θ | R = r

)

= ln k + m lnα + mα lnβ − (α + 1)m∑i=1

ln(xi − θ)

− βαm∑i=1

(xi − θ)−α +m∑i=1

ri ln

[1− exp

(−[

β

xi − θ

]α)],

(5)

where k is defined in (3). Now, suppose that an individualunit being removed from the life test is independent of theothers but with the same probability p. Then, the numberof units removed at each failure time follows a binomialdistribution such that

P(R1 = r1) =⎛⎝n−m

r1

⎞⎠pri(1− p

)n−m−r1 , (6)

where 0 ≤ r1 ≤ n

P(Ri = ri | Ri−1 = ri−1, . . . ,R1 = r1)

=

⎛⎜⎜⎝n−m−

i−1∑k=1

rk

ri

⎞⎟⎟⎠× pri

(1− p

)n−m−∑i−1k=1 rk ,

(7)

where 0 ≤ ri ≤ n −m −∑ik=1 rk, i = 2, . . . ,m − 1. Suppose

further that Ri is independent of Xi. Then, the Log-likelihoodfunction can be expressed as

L∗(α,β, θ

) = L∗(α,β, θ | R = r

)P(R = r), (8)

where

P(R = r)

= P(Rm−1 = rm−1 | Rm−2 = rm−2, . . . ,R1 = r1)

· · ·P(R2 = r2 | R1 = r1)P(R1 = r1),

(9)

P(R = r)

= (n−m)!∏m−1i=1 ri!

(n−m−∑m−1

i=1 ri)

!

× p∑m−1

i=1 ri(1− p

)(m−1)(n−m)−∑m−1i=1 (m−i)ri .

(10)

International Journal of Quality, Statistics, and Reliability 3

Independently, the MLE of parameter p can be obtained bymaximizing (10). Thus, we find immediately

p =∑m−1

i=1 ri(m− 1)(n−m)−∑m−1

i=1 (m− i− 1)ri. (11)

Note that P(R = r) does not depend on the parameters α, β,and θ, and hence the MLE’s of the parameters can be derivedfrom (5) by differentiating with respect to α, β, and θ andequating to zero, in this case we have

m−m∑i=1

[β

xi − θ

]α

+m∑i=1

ri[β/(xi − θ)

]α exp(−[β/(xi − θ)

]α)1− exp

(−[β/(xi − θ)

]α) = 0,

m∑i=1

⎛⎝1 + α

α

[β

xi − θ

]−[

β

xi − θ

](α+1)

+ri[β/(xi − θ)

](α+1) exp(−[β/(xi − θ)

]α)1− exp

(−[β/(xi − θ)

]α)⎞⎠ = 0,

m

α+ m lnβ

+m∑i=1

(−[

β

xi − θ

]α

ln

[β

xi − θ

]

+ri ln

[β/(xi − θ)

]exp

(−[β/(xi − θ)

]α)1− exp

(−[β/(xi − θ)

]α)⎞⎠ = 0,

(12)

since (12) cannot be solved analytically for estimators α, β,

and θ. Hence, in this paper, the numerical solution is used tosolve the problem.

4. Some Further Results

In this section, we are going to derive the sampling distri-butions of the MLE’s and obtain the confidence intervalsfor the parameters [13–15]. In addition, we will obtain theMLE and confidence interval for the percentile of failure timedistribution.

4.1. Distributions and Confidence Intervals for the Parameters.Let X1 < X2 < · · · < Xm−1 < Xm denote a progressivelyType II censored sample from a three-parameter Frechetdistribution with censored scheme R = (r1, . . . , rm). Let Yi =[(xi − θ)/β]−α, i = 1, . . . ,m. It can be seen that, Z1 < Z2 <· · · < Zm is a progressively Type II censored sample from

an exponential distribution with mean 1. Let us consider thefollowing transformation

Z1 = nY1

Z2 = (n− r1 − 1)(Y2 − Y1)

......

Zm = (n− r1 − · · · − rm−1 −m + 1)(Ym − Ym−1).

(13)

Thomas and Wilson [16] showed that the generalized spac-ings Z1,Z2, . . . ,Zm as defined in the previous equation (13)are all independent and identically distributed as standardexponential. V = 2Z1 = 2n[(xi − θ)/β]−α has a Chi-squaredistribution with 2 degrees of freedom. We can also write the

numerator of β as the sum of m− 1 independent generalizedspacings, that is, 2

∑mi=1(ri+1)Yi−nY1 = 2

∑mi=2 Zi. Therefore,

we can find that, conditionally on a fixed set of R =(r1, . . . , rm), U = 2

∑mi=1(ri + 1)Yi − nY1 has a Chi-square

distribution with 2m − 2 degrees of freedom. It is also easilyseen that V and U are independent. Let

T1 = U(m− 1)V

=∑m

i=1(ri + 1)Yi − nY1

n(m− 1)Y1,

T2 = U + V = 2m∑i=1

(ri + 1)Yi.

(14)

It is easy to show that, T1 has an F distribution with2m − 2 and 2 degrees of freedom, and T2 has a Chi-squaredistribution with 2m degrees of freedom. Furthermore, T1

and T2 are independent.

Theorem 1. Suppose that w1 < w2 < · · · < wm. Let

T1(w) =∑m

i=1(ri + 1)(wi − θ)−α − n(w1 − θ)−α

n(m− 1)(w1 − θ)−α. (15)

Then, T1(w) is strictly increasing in w for any w > 0,furthermore, if t > 0, the equation T1(w) = t has uniquesolution for any w > 0.

Suppose that, Xi, i = 1, . . . ,m are order statistics of aprogressively Type II censored sample of size n from theFrechet distribution, with censoring scheme (r1, . . . , rm) thea 100(1− φ) confidence interval for α is

Ψ(X1, . . . ,Xm,Fφ/2(2m− 2, 2)

)

< α < Ψ(X1, . . . ,Xm,F1−φ/2(2m− 2, 2)

).

(16)

4 International Journal of Quality, Statistics, and Reliability

Such that,

1− φ

= P[Fφ/2(2m− 2, 2) < T1 < F1−φ/2(2m− 2, 2)

]

= P[Fφ/2(2m− 2, 2)

<

∑mi=1(ri + 1)(xi − θ)−α − n(x1 − θ)−α

n(m− 1)(x1 − θ)−α

< F1−φ/2(2m− 2, 2)].

(17)

Furthermore, the 100(1 − φ) joint confidence region for αand β is determined by the following inequalities:

Ψ(X1, . . . ,Xm,F(1+

√1−φ)/2(2m− 2, 2)

),

< α < Ψ(X1, . . . ,Xm,F(1−√1−φ)/2(2m− 2, 2)

),

(18)

χ2(1+√

1+φ)/2(2m)[2∑m

i=1(ri+1)(xi−θ)α]1/α <β<

χ2(1+√

1−φ)/2(2m)[2∑m

i=1(ri+1)(xi−θ)α]1/α .

(19)

Such that,

1− φ =√

1− φ√

1− φ

=[PF(1+

√1−φ)/2(2m− 2, 2) < T1

< F(1−√1−φ)/2(2m− 2, 2)]

× P[χ2(1+√

1−φ)/2(2m) < T2 < χ2(1−√1−φ)/2(2m)

].

(20)

4.2. Confidence Interval of xF . In reliability analysis [17,18], we are not only interested in making inference aboutparameter but also interested in deriving inference aboutpercentiles of the failure time distribution. Let xF be the100F th percentile of the failure time distribution. One canobtain xF by solving FX(xF) = p where FX(·) is as givenin (2). Then it easy to see that xF = θ − β(log p)−1/α.

Consequently, the MLE of xF is given by xF = θ −β(log p)−1/α. Confidence interval for xF can be derived by

using the pivotal quantity UF = [(xF − θ)/β]−α

if (u f ,1−φ)−α

is the (1− φ)th percentile of UF , then

1− φ = P[UF ≤ uF,1−φ

]= P

[xF ≥ θ − βuF,1−φ

]. (21)

Hence, θ − βuF,1−φ is a lower confidence limit for xF withconfidence coefficient 1− φ. We can rewrite (21) as

1− φ = P[V

2n+ λU ≤ − ln p

], (22)

such that, V ∼ χ2(2), U ∼ χ2(2m− 2), and λ = −uF,1−φ/2m.In addition, V and U are independent, which is discussedin the previous subsection. Hence, we need to find the valueof λ by using solving (22). There are several methods forcomputing λ, see Engelhardt and Bain [19].

5. Conclusions

We develop some results on a three-parameter Frechet dis-tribution when progressive Type II censoring with binomialremovals is performed. We derive the MLEs and confidencefor the parameters. The MLE and confidence interval forthe percentiles of failure time distribution are obtained. Inpractice, it is often useful to have an idea of the durationof a life test. Therefore, it is important to compute theexpected time required to complete a life test. In thecase of progressively type II-censored sampling plan withbinomial removals, one can obtain this information bycalculating the expectation of the mth order statistic. Infact, we believe that the value of removal probability p isvery important when we compare the expected test timesof progressive Type II censoring with binomial removalsand complete sampling plan. In addition, the removalprobability p may not be fixed for each stage. Such beliefis not discussed in this paper and we will be investigated inthe future.

List of Symbols

f (x): Probability density functionF(x): Cumulative distribution functionα, β, θ: Three-parameter Frechet distributionR: Number of removalsp: Removal probabilityχ2: Chi-squareF: F-distribution100(1− φ): Confidence intervalXF : 100-th percentile of the failure time

distribution.

References

[1] S. Nadarajah and S. Kotz, “The Exponentiated Frechet Dis-tribution,” Interstat Electronic Journal, 2003, http://interstat.statjournals.net/YEAR/2003/articles/.

[2] A. M. Abd-Elfattah and A. M. Omima, “Estimation of theunknown parameters of the generalized Frechet distribution,”Journal of Applied Sciences Research, vol. 5, no. 10, pp. 1398–1408, 2009.

[3] S. Kotz and S. Nadarajah, Extreme Value Distributions: Theoryand Applications, Imperial College Press, London, UK, 2000.

[4] N. R. Mann, R. E. Schafer, and N. D. Singpurwalla, Methodsfor Statistical Analysis of Reliability and Life Data, John Wiley& Sons, New York, NY, USA, 1974.

[5] J. F. Lawless, ”Statistical Methods and Methods for LifetimeData, John Wiley & Sons, New York, NY, USA, 1982.

[6] W. Q. Meeker and L. A. Escobar, Statistical Methods forReliability Data, John Wiley & Sons, New York, NY, USA, 1998.

[7] N. Balakrishnan and R. Aggarwala, Progressive Censoring—Theory, Moethods and Applications, Birkhaauser, Boston, Mass,USA, 2000.

[8] A. C. Cohen, “Progressively censored samples in life testing,”Technometrics, vol. 5, pp. 327–339, 1976.

[9] N. R. Mann, “Best liner invariant estimation for Weibullparameters under progressive censoring,” Technometrics, vol.13, pp. 521–533, 1971.

International Journal of Quality, Statistics, and Reliability 5

[10] R. Viveros and N. Balakrishnan, “Interval estemation ofparameters of life from progressivve censoring data,” Techno-metrics, vol. 36, pp. 84–91, 1994.

[11] H. K. Yuen and S. K. Tse, “Parameters estimation for weibulldistributed lifetimes under progressive censoring with randomremovals,” Journal of Statistical Computation and Simulation,vol. 55, no. 1-2, pp. 57–71, 1996.

[12] S. K. Tae, C. Yang, and H. K. Yuen, “Statistical analysis ofWeibull distributed lifetime data under Type II prograssivecensoring with binomial removals,” Journal of Applied Statis-tics, vol. 27, pp. 1033–1043, 2000.

[13] K. Alakus, “Confidence intervals estimation for survivalfunction in weibull proportional hazards regression based oncensored survival time data,” Scientific Research and Essays, vol.5, no. 13, pp. 1589–1594, 2010.

[14] M. Maswadah, “Conditional confidence interval estimationfor the inverse weibull distribution based on censored gener-alized order statistics,” Journal of Statistical Computation andSimulation, vol. 73, no. 12, pp. 887–898, 2003.

[15] J. A. Griggs and Y. Zhang, “Determining the confidenceintervals of Weibull parameters estimated using a more preciseprobability estimator,” Journal of Materials Science Letters, vol.22, no. 24, pp. 1771–1773, 2003.

[16] D. R. Thomas and W. M. Wilson, “Linear order statisticestimation for the two parameter Weibull and extreme valuedistribution from type -II progressively censored samples,”Technometrics, vol. 14, pp. 679–691, 1972.

[17] M. Han, “Estimation of failure probability and its applicationsin lifetime data analysis,” International Journal of Quality,Statistics, and Reliability, vol. 2011, Article ID 719534, 6 pages,2011.

[18] S. Loehnert, “About statistical analysis of qualitative surveydata,” International Journal of Quality, Statistics, and Reliabil-ity, vol. 2010, Article ID 849043, 12 pages, 2010.

[19] M. Engelhardt and L. J. Bain, “Tolerance limits and confidencelimits on reliability for the two parameter exponential distri-bution,” Technometrics, vol. 20, pp. 37–39, 1978.

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