role of equilibrium distribution in reliability studies

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ROLE OF EQUILIBRIUM DISTRIBUTION IN RELIABILITY STUDIES

Ramesh C. Gupta

Probability in the Engineering and Informational Sciences / Volume 21 / Issue 02 / April 2007, pp 315 334DOI: 10.1017/S0269964807070192, Published online: 27 February 2007

Link to this article: http://journals.cambridge.org/abstract_S0269964807070192

How to cite this article:Ramesh C. Gupta (2007). ROLE OF EQUILIBRIUM DISTRIBUTION IN RELIABILITY STUDIES. Probability in the Engineering and Informational Sciences, 21, pp 315334 doi:10.1017/S0269964807070192

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ROLE OF EQUILIBRIUMDISTRIBUTION IN

RELIABILITY STUDIES

RAAAMMMEEESSSHHH C. GUUUPPPTTTAAADepartment of Mathematics and Statistics

University of MaineOrono, ME 04469-5752

E-mail: [email protected]

The equilibrium distribution arises as the limiting distribution of the forward recur-rence time in a renewal process+ The purpose of this article is to study the relation-ships between the equilibrium distributions ~including its higher derivates! and theoriginal distributions+ Some stochastic order relations and the relations between theiraging properties are investigated and some applications in the field of insurance andfinancial investments are given+ In addition, the relation between the equilibrium dis-tribution of a series system and the series system of equilibrium distribution is inves-tigated+Bivariate equilibrium distribution whose reliability properties are consistentwith those of the univariate equilibrium distribution is defined+

1. INTRODUCTION

Let X be a continuous random variable with probability density function ~p+d+f+!f ~x!, distribution function F~x!, and survival function OF~x!�1 � F~x!+We definethe p+d+f+ f *~x!as

f *~x! �OF~x!

µ, x � 0, (1.1)

where µ � E~X ! � `+ Then f *~x! is called the p+d+f+ of an equilibrium distributionor induced distribution+

The above distribution arises as the limiting distribution of the forward recur-rence time in a renewal process+ It also arises as the marginal distribution of W1,where the joint p+d+f+ of ~W1,W2! is given by

Probability in the Engineering and Informational Sciences, 21, 2007, 315–334+ Printed in the U+S+A+DOI: 10+10170S0269964807070192

© 2007 Cambridge University Press 0269-9648007 $25+00 315

g~w1,w2 ! �f ~w2 !

µ, 0 � w1 � w2 � `

� 0, elsewhere;

see Brown @10# +Note that, in this case, the p+d+f+ of W2 is given by the length-biased version of

the original distribution as

fW2~w2 ! �

w2 f ~w2 !

µ, 0 � w2 � `+

The equilibrium distribution ~1+1! is intimately connected to its parent distri-bution and many of the reliability properties of the original distribution can be eas-ily studied by means of the properties of the equilibrium distribution+

The purpose of this article is to study the relationships between ~1+1! ~includ-ing its higher derivatives! and the original distribution+ Some stochastic order rela-tions and the relations between their aging properties are investigated and someapplications in the field of insurance and financial investments are given+ This isprimarily a review article+ However, the examples in Section 6 are new+

The organization of this article is as follows+ In Section 2 we present somedefinitions and background material encountered in reliability studies, includingsome criteria of aging and their relationships+ Some definitions of stochastic orderrelations and their relationships are also provided+ Section 3 deals with higher-order equilibrium distributions and stop loss moments+ In Section 4 aging proper-ties of equilibrium distributions and their stochastic ordering with the originaldistribution are explored+ In Section 5 the relation between the equilibrium distri-bution of a series system and a series system of equilibrium distributions, consist-ing of two components, is investigated+ Section 6 contains the bivariate equilibriumdistribution along with two examples+ Finally, in Section 7 we provide some con-clusions and comments+

2. DEFINITIONS AND BACKGROUND

Let X be a continuous positive random variable representing a survival time with anabsolutely continuous distribution function F~t !, survival function OF~t !�1 � F~t !,and cumulative hazard rate L~t !� �ln OF~t ! with OF~0!� 1+Wherever necessary, itwill be assumed that rF~t !� L'~t ! is the hazard rate corresponding to F~t !+ A keyrole in this article will be played by the mean residual life function ~MRLF! µF~t !defined as

µF ~t ! � E~X � t 6X � t !�

�t

`

OF~x! dx

OF~t !, t � 0+

316 R. C. Gupta

It will be assumed that µF~0!� E~X ! � `+ When discussing the variance ofthe residual lifetime X � t 6X � t, it will be assumed that E~X 2!�`+ The varianceresidual life function ~VRLF! is defined as

sF2~t ! � Var~X � t 6X � t !

�2

OF~t !�

t

`�x

`

OF~ y! dy dx � µF2 ~t !, t � 0+

We refer to Gupta and Kirmani @17,18# , Launer @28# , and Gupta, Kirmani, andLauner @19# for details about the MRLF and the VRLF+

The above-defined functions highlight different aspects of survival and re-sidual life distributions+The hazard rate and the mean residual life function arerelated by

rF ~t ! �1 � µF

' ~t !

µF ~t !+ (2.1)

Further, the hazard rate, the MRLF, and the VRLF are tied together by therelation

d

dtsF

2~t ! � rF ~t !$sF2~t !� µF

2 ~t !%; (2.2)

see Gupta @16# +It is well known that rF~t ! determines the distribution function uniquely and,

hence, µF~t ! also characterizes the distribution+ Additionally, OF~t ! and µF~t ! areconnected by

OF~t ! �µF ~0!

µF ~t !exp��

0

t dx

µF ~x!� + (2.3)

Thus, rF~t !, µF~t !, and OF~t ! are equivalent in the sense that given one of them, theother two can be determined+Hence, in the analysis of survival data, one sometimesestimates rF~t ! or µF~t ! instead of OF~t !, according to the convenience of the pro-cedure available+

In addition to the above functions, the residual coefficient of variation is givenby gF~t !�sF~t !0µF~t !+ Further, the MRLF, the VRLF, and the residual coefficientof variation are connected by the relation

d

dtsF

2~t ! � µF ~t !~1 � µF' ~t !!~gF

2~t !� 1!; (2.4)

see Gupta @16# +We now describe briefly some aging classes of life distributions and their

relationships

EQUILIBRIUM DISTRIBUTION IN RELIABILITY 317

2.1. Some Criteria of Aging for the RLF

In this section we review some of the aging criteria and their relationships+We alsodescribe how the aging properties of the original distribution are transformed intothe aging properties of the residual life+

Let X be a continuous positive random variable representing the life of acomponent+ Let F be the cumulative distribution function of X and OF ~x! �1 � F~x! be the reliability function or the survival function of X+ Then Ft~x! �P~X � x � t 6X � t ! is the survival function of a unit of age t+ Evidently, anystudy of the phenomenon of aging should be based on Ft~x! and functions relatedto this+ Thus, the following hold:

1+ F is said to be PF2 if ln f ~x! is concave, where f ~{! is the density corre-sponding to F~{!+

2+ F is said to have increasing ~decreasing! failure rate @IFR ~DFR!# if Ft~x!�OF~x � t !0 OF~t ! is decreasing ~increasing! in t+ If F is absolutely continuous

with density f, then F is in the IFR ~DFR! class if rF~t !� f ~t !0 OF~t ! is increas-ing ~decreasing! in t+

3+ F is said to have increasing ~decreasing! failure rate average @IFRA ~DFRA!#if *0

t rF ~x! dx0t is increasing ~decreasing!+4+ F is said to have new better ~worse! than used @NBU ~NWU!# if OFt~x!� ~�!OF~x! for x � 0 and t � 0+

5+ F is said to have decreasing ~increasing!mean residual life @DMRL ~IMRL!#if the mean residual life µF~t !� *t

` OF~x! dx0 OF~t ! is decreasing ~increasing!assuming that the mean µF~0! exists+

6+ F is said to have new better ~worse! than used in expectation @NBUE~NWUE!# if µF~t ! � ~�! µF~0! for all t � 0+

7+ F is said to have decreasing variance residual life ~increasing variance resid-ual life! @DVRL ~IVRL!# if sF

2~t ! is decreasing ~increasing!+

The chain of implications among these classes of distributions is

PF2 n IFR n IFRA n NBU

⇓ ⇓

DMRL n NBUE+

⇓

DVRL

The reverse implications are not true; for counterexamples, see Bryson andSiddiqui @11# + Some extensions of these classes of distributions are containedin Klefsjo @26,27# , Shaked @32# , Singh and Deshpande @35# , Deshpande, Kochar,and Singh @12#, Basu and Ebrahimi @5,6#,Abouammoh @1#,Abouammoh and Ahmad@2# , and Loh @30# +

318 R. C. Gupta

2.2. Stochastic Order Relations

Let X and Y be nonnegative absolutely continuous random variables with densityfunctions f ~x! and g~x! and survival functions OF~x! and OG~x!, respectively+ Then,we have the following:

1+ X is said to be smaller than Y in the likelihood ratio ordering, written asX �LR Y, if f ~x!0g~x! is nonincreasing in x+

2+ X is said to be smaller than Y in the failure ~hazard! rate ordering, written asX �FR Y, if rF~x! � rG~x! for all x+

3+ X is said to be smaller than Y in the stochastic ordering, written as X �st Y,if OF~x! � OG~x! for all x+

4+ X is said to be smaller than Y in the mean residual life ordering, written asX �MRL Y, if µF~x! � µG~x! for all x+ Deshpande, Singh, Bagai, and Jain@13# show that X �MRL Y if and only if *x

` OF~u! du0*x` OG~u! du is decreas-

ing in x+5+ X is said to be smaller than Y in the increasing convex order, written as

X �icx Y, if *x` OF~u! du � *x

` OG~u! du for all x+ Note that in the literature theincreasing convex order has also been called stop loss ordering ~Hesselager@22# ! and ST2 ordering; see Belzunce, Candel, and Ruiz @7# and Shaked andShanthikumar @33# + For some generalized variability ordering, see Zarek@39# , Li and Zhu @29# , and Bhattacharjee and Sethuraman @8# +

6+ X is said to be smaller than Y in the variance residual life ordering, writtenas X �VRL Y, if sF

2~x! � sG2~x! for all x+

7+ X is said to be smaller than Y in the Laplace transform ordering, written asX �LT Y, if E~e�sX!� E~e�sY!+ Shaked and Shanthikumar @33# showed thatthis is equivalent to *0

` e�sx OF~x! dx � *0` e�sx OG~x! dx+

8+ X is said to be smaller than Y in the moment generating function ordering,written as X �MGF Y, if E~esX! � E~esY!+

It is well known that

X �LR Yn X �FR Yn X �MRL Y n X �VRL Y

⇓ ⇓

X �st Y X �icx Y;

see Shaked and Shanthikumar @33# +

3. HIGHER-ORDER EQUILIBRIUM DISTRIBUTIONS AND STOPLOSS MOMENTS

We define the sequence of induced distributions as follows+ Let OF~x! be the sur-vival function of a nonnegative random variable X with MRLF given by µF~t !+Wenow define a sequence $ OF1, OF2, + + +% of survival functions induced by OF as


Fn~t ! �

�t

`

OFn�1~u! du

µn�1

, (3.1)

where µn is the mean of the distribution Fn+ Assume that E~Xn!� ` for the largestn used above so that µ1, µ2, + + + , µn will all be finite+ In what follows, we will denoteby X @0# � X, corresponding to the survival function OF0 � OF, the original randomvariable and X @1#, X @2#, + + + corresponding to OF1, OF2, + + + +

We now present the following result+

Theorem 3.1: Let f be a convex function. Then

EF @f~X � t !6X � t #� f~0!� EF @~X � t !6X � t #EF1@f '~X � t !6X � t # + (3.2)

Proof:

EF @f~X � t !6X � t #

�

��t

`

f~x � t ! d OF~x!

OF~t !

� f~0!��

t

`

f '~x � t ! OF~x! dx

OF~t !

� f~0!��t

`

f '~x � t ! �OF~x!0µ

�t

`

~ OF~x!0µ! dx ��

t

`

OF~x! dx

OF~t !dx

� f~0!� EF @~X � t !6X � t #�t

`

f '~x � t !f1~x!

F1~x!dx,

where f1~x!� OF~x!0µ is the p+d+f+ of the first induced distribution and F1~x! is itssurvival function+ Thus,

EF @f~X � t !6X � t #� f~0!� EF @~X � t !6X � t #EF1@f '~X � t !6X � t # + �

Particular Cases:

EF @~X � t !2 6X � t #� 2µF ~t !µF1~t ! (3.3)

and, in general,

EF @~X � t !k 6X � t #� k!)j�0

k�1

µj ~t !, (3.4)

320 R. C. Gupta

where µ0~t !� µ~t !� MRLF of F and µj~t ! is the MRLF of the j th induced distri-bution; see also Stein and Dattero @36# + From this, it is clear that

EF ~Xk 6X � t !� k!)

j�0

k�1

µj , (3.5)

where µj is the mean of the j th induced distribution+

3.1. Stop Loss Transform

Definition 3.1: The function pX~t ! � E @~X � t !�# is called the stop loss trans-form of X, where ~X � t !� � Max~X � t,0! represents the amount by which Xexceeds the threshold t.

The kth stop loss moment is given by

Rk~t ! � E @~X � t !�k #��

t

`

~x � t !k dF~x!; (3.6)

see Willmot, Drekic, and Cai @38# and Hesselager, Wang, and Willmot @23# + Notethat Rk~t ! has also been called the kth partial moment in the literature; see Guptaand Gupta @15# +We now present the following result+

Theorem 3.2: The survival function of the kth equilibrium distribution is given by

OFk~t ! � E @~X � t !�k #0E~X k !+ (3.7)

Proof:

E @~X � t !�k #0E~X k !

��t

`

~x � t !k dF~x!0E~X k !

��t

`

k~x � t !k�1 OF~x! dx0E~X k !

� kµ�t

`

~x � t !k�1 f1~x! dx0E~X k !

�kµ

E~X k !�

t

`

~k � 1!~x � t !k�2 OF1~x! dx

�k~k � 1!µµ1

E~X k !�

t

`

~x � t !k�2 f2~x! dx+


Proceeding in this way, we get

E @~X � t !�k #0E~X k ! �

k!Pj�0k�1 µj

E~X k !OFk~t !

� OFk~t !+ �

We now show that Rk~t ! determines the distribution function uniquely+

Theorem 3.3: Under the above-stated conditions,

OF~t ! �~�1!k

k!Rk~k!~t !, (3.8)

where Rk~k!~t ! denotes the kth derivative of Rk~t ! .

Proof: Applying Taylor’s theorem to F~x!, we get

F~x! � 1 �~�1!k

k!�

t

`

~x � t !kF ~k! ~x! dF~x!

� 1 �~�1!k

k!� d

dt�k�

0

`

uk f ~t � u! du

� 1 �~�1!k

k!� d

dt�k�

t

`

~x � t !k dF~x!

� 1 �~�1!k

k!Rk~k!~t !+ (3.9)

This gives the desired result+ �

Remark 3.1: For other proofs, see Gupta and Gupta @15# and Navarro, Franco, andRuiz @31# +

4. AGING PROPERTIES OF EQUILIBRIUM DISTRIBUTIONS ANDSOME ORDER RELATIONS

It is well known that the failure rate of F1 is the reciprocal of the MRLF of F and,in general, the failure rate of Fi ~i � 1,2, + + +! is the reciprocal of the MRLF of Fi�1

~i � 1,2, + + +!+ This means that Fi is IFR is equivalent to Fi�1 is DMRL+ In the fol-lowing, we show that Fi�1 is DMRL is equivalent to Fi�2 is DVRL+

Theorem 4.1: Fi�1 is DMRL is equivalent to Fi�2 is DVRL ~i � 2,3, + + +! .

Proof: It is enough to prove the result for i � 2+ We have

E~~X � t !2 6X � t !

E~~X � t !6X � t !� 2µF1

~t !+ (4.1)

322 R. C. Gupta

Also,

µF1~t ! �

sF2~t !� µF

2 ~t !

µF ~t !

�1

2µF ~t !~gF

2~t !� 1!+

This gives

d

dt@E~~X � t !2 6X � t !#0E~X � t 6X � t !

� 2µF1

' ~t !

� 2~rF1~t !µF1

~t !� 1!

� 2�µF1~t !

µF ~t !� 1�

� 2� 1

2~gF

2~t !� 1 � 1!� gF

2~t !� 1+

This proves the result; see also Gupta et al+ @19# , Singh @34# , Fagiuoli and Pellerey@14# , and Willmot @37# + �

Corollary 4.1: Suppose F is a strictly increasing life distribution. Then F is DVRL(IVRL) if and only if the induced distribution corresponding to F is DMRL.

Remark 4.1: The restriction to strictly increasing F in Theorem 4+1 and Corollary4+1 cannot be relaxed+ This can be seen from Example 1 of Gupta et al+ @19# , whereF is IVRL but the function E~~X � t !2 6X � t !0E~~X � t !6X � t ! is not increasingin the neighborhood of t � 1

2_ +

Remark 4.2: It is also clear from the above derivation that

µF1~t !

µF ~t !�

1

2~1 � gF

2~t !!+

From the above, we can state the following+

Theorem 4.2: Suppose F is a strictly increasing life distribution. Then F is DVRL(IVRL) if and only if µF1

~t ! � ~�! µF~t ! .

Remark 4.3: The above result was also noticed by Deshpande et al+ @13# and Abou-ammoh, Kanjo, and Khalique @3# +


It is also clear from the above that µF1~t !� µF~t ! if and only if F has an expo-

nential distribution+The following theorem addresses the stochastic comparisons of X @1# and X+

Theorem 4.3: Suppose X is a nonnegative random variable with E~X !�`. ThenX is NBUE (NWUE) if X @1# �st ~�st ! X.

Proof: The survival function of X @1# is given by

OFX @1# ~t ! ��t

`

OF~x! dx0µ

� OFX ~t ! OFX ~t !0µ+

This gives

OFX @1# ~t !

OFX ~t !�

µF ~t !

µ+

Thus,

OFX @1# ~t ! � OFX ~t ! if and only if X is NBUE+

Similarly, OFX @1# ~t ! � OFX~t ! if and only if X is NWUE+ �

We now present the following result dealing with the hazard rate comparisonof X @1# and X+

Theorem 4.4: Suppose X is a nonnegative random variable with E~X !�`. ThenX is IMRL (DMRL) if and only if X �FR ~�FR! X @1#.

Proof:

X � X @1# m OFX @1# ~t !0 OFX ~t ! is increasing

mOFX @1# ~v!

OFX ~v!�OFX @1# ~u!

OFX ~u!if u � v

m OFX @1# ~v! OFX ~u!� OFX @1# ~u! OFX ~v!

m �v

` OF~t !

µdt OFX ~u!� �

u

` OF~t !

µdt OFX ~v!

m

�v

`

OF~t ! dt

OFX ~v!�

�u

`

OF~t ! dt

OFX ~u!

m µX ~v!� µX ~u!

m X is IMRL+ �

324 R. C. Gupta

The following result deals with the Laplace ordering of two random variablesand their corresponding equilibrium distributions+

Theorem 4.5: Let X and Y be two nonnegative random variables with E~X ! �E~Y ! . Then

X �LT Ym X @1# �LT Y @1#+

Proof: We have

LX @1# ~t ! � t�0

`

e�tx��0

x OFX ~ y!

µdy� dx

�t

E~X !�

0

`��y

`

e�tx dx� OFX ~ y! dy

�1

E~X !�

0

`

e�ty OFX ~ y! dy+

Now

X �LT Ym �0

`

e�tx OFX ~x! dx � �0

`

e�ty OFY ~ y! dy

m LX @1# ~t !� LY @1# ~t ! since E~X !� E~Y !+ �

The following two results deal with the stop loss ordering of two random vari-ables and their corresponding equilibrium variables+

Theorem 4.6: Let X and Y be two nonnegative random variables with E~X ! �E~Y ! . Then

X �SL Ym X @1# �st Y @1#+

Proof:

X @1# �st Y @1# mpX ~t !

E~X !�pY ~t !

E~Y !for all t � 0

m pX ~t !� pY ~t ! since E~X !� E~Y !

m X �SL Y+ �


Theorem 4.7: Let X and Y be two nonnegative random variables with finite means.Then

X �HR Ym X @1# �LR Y @1#+

Proof:

X @1# �LR Y @1#

mfX @1% ~t !

fY @1% ~t !is decreasing

mfX @1% ~u!

fY @1% ~u!�

fX @1% ~v!

fY @1% ~v!if u � v

mOFX ~u!

OFY ~u!�OFX ~v!

OFY ~v!

mOFX ~t !

OFY ~t !is decreasing

m X �HR Y+ �

We now define the convex ordering as follows+

Definition 4.1: X �icx Y if E~ f ~X !! � E~ f ~Y !! for all increasing convex func-tions. This is equivalent to

X �icx Y if OFX ~t ! � OFY ~t ! for all t � 0.

Following the above concept, we define a similar ordering between the sthequilibrium as follows+

Definition 4.2: X �s�icx Y if OFX~s!~t ! � OFY

~s!~t ! for all t � 0, where OFX~s!~t ! and

OFY~s!~t ! are the survival functions of their sth equilibrium distributions; see Klar

and Muller [25].

Comparing the original variable and its equilibrium variable in the above order-ing, we have

X �s�icx X @s# for s � 1 if E~X � t !�s�1 � E~X @s# � t !�

s�1 +

This is equivalent to

E~X � t !�s

E~X � t !�s�1

� µs+

326 R. C. Gupta

Denoting

Mn~t ! �E~X � t !�

n

nE~X � t !�n�1,

the above inequality can be written as

Ms~t ! � µ;

see Stein and Dattero @36# +

5. EQUILIBRIUM DISTRIBUTION OF A SERIES SYSTEM

Let X1, X2, + + + , Xn be independent random variables with survival functions OFX1,

OFX2, + + + , OFXn

, respectively+ When the components are arranged in a series system,we observe X1 ∧ X2 ∧ {{{ ∧ Xn � Min~X1, X2, + + + , Xn!+ In this section, we will com-pare two systems whose components are the nth equilibrium distributions of X andY and the nth equilibrium distribution of a system having components X and Y+More precisely, we have the following result due to Bon and Illayk @9# +

Theorem 5.1: Let X and Y be independent nonnegative random variables. LetX @k# and Y @k# be respectively the independent equilibrium variables of X and Y oforder k. If X and Y have the DMRL property and if the nth moments of X and Y exist,then

X @n# ∧ Y @n# �LR ~X ∧ y!@n#,

where ~X ∧ y!@n# is the equilibrium variable of X ∧ Y of order n.

Proof: For n � 1, X @1# and Y @1# are independent and the p+d+f+ of X @1# ∧ Y @1# isgiven by

fX @1#∧Y @1# ~t ! � fX @1# ~t ! OFY @1# ~t !� fY @1# ~t ! OFX @1# ~t !+

Also, the p+d+f+ of ~X ∧ Y !@1# is

f~X∧Y !@1# ~t ! �OFX ~t ! OFY ~t !

E~X ∧ Y !+

These give

fX @1#∧Y @1# ~t !

f~X∧Y !@1# ~t !�

E~X ∧ Y !@µX ~t !� µY ~t !#

E~X !E~Y !+

Since X and Y have the DMRL property, the right-hand side of the above equa-tion is decreasing and, hence, X @1# ∧ Y @1# �LR ~X ∧ Y !@1#+


For the general case, we proceed by induction and assume that the result is trueof k � n+ It can be verified that

fX @n#∧Y @n# ~t !

f~X∧Y !@n# ~t !

� E~X ∧ Y !@n�1#� OFX @n�1# ~t ! OFY @n# ~t !

E~X @n�1# ! OF~X∧Y !@n�1# ~t !�

OFY @n�1# ~t ! OFX @n# ~t !

E~Y @n�1# ! OF~X∧Y !@n�1# ~t ! ,� A~t !� B~t !,

where

A~t ! �OFX @n�1# ~t ! OFY @n# ~t !

E~X @n�1# ! OF~X∧Y !@n�1# ~t !,

and likewise for B~t !+ We now show that A~t ! and B~t ! are decreasing functionsof t+

Since Y has the DMRL property, all of its equilibrium variables have the IFRproperty and also the DMRL property+ This means that Y @n# �HR Y @n�1#+ Thus,

X @n�1# ∧ Y @n# �HR X @n�1# ∧ Y @n�1#+

Also, the induction hypothesis implies that

X @n�1# ∧ Y @n�1# �HR ~X ∧ Y !@n�1#+

Therefore, X @n�1# ∧ Y @n# �HR ~X ∧ Y !@n�1#+ This implies that A~t ! is decreasing+Similary, B~t ! is decreasing and the proof is complete+ �

Remark 5.1: For the lower bound on the nth moment of a series system, see theabove remark+

6. BIVARIATE EQUILIBRIUM DISTRIBUTION

In order to define bivariate equilibrium distribution, we first review the bivariatehazard rates and bivariate MRLF as follows+ Let ~X1, X2! be a nonnegative bivar-iate random variable with survival function OF~x1, x2!+The hazard components of~X1, X2! are given by h1~x1, x2! and h2~x1, x2!, where

hi ~x1, x2 ! �]

]xi

@�ln OF~x1, x2 !# , i � 1,2,

328 R. C. Gupta

and the MRLF of ~X1, X2! is given by ~µ1~x1, x2!, µ2~x1, x2!!, where

µ1~x1, x2 ! � E~X1 � x16X1 � x1, X2 � x2 !

�1

OF~x1, x2 !�

x1

`

OF~x1, x2 ! dx1,

and likewise for µ2~x1, x2!+ The hazard components and the MRLF components arerelated by

hi ~x1, x2 ! �

1 �]

]xi

µi ~x1, x2 !

µi ~x1, x2 !, i � 1,2+

Keeping these properties in mind, Gupta and Sankaran @21# defined the bivar-iate equilibrium random variable ~Y1,Y2! by means of the conditional distributionas follows+

The p+d+f+ of Y1 given Y2 � y2 is

g~ y16Y2 � y2 !�P~X1 � y16X2 � y2 !

E~X16X2 � y2 !+ (6.1)

Likewise, the p+d+f+ of Y2 given Y1 � y1 is

g~ y2 6Y1 � y1!�P~X2 � y2 6X1 � y1!

E~X2 6X1 � y1!+ (6.2)

It can be easily seen that the marginal distributions of Y1 and Y2 coincide with theequilibrium distributions of X1 and X2, respectively+ Following the above scheme,one can easily define equilibrium distributions in higher dimensions+

We now present two examples+

Example 6.1: Gumbel’s bivariate exponential distribution with survival function

OF~x1, x2 ! � exp~�ax1 � bx2 � cx1 x2 !, x1, x2 � 0,a,b, c � 0+

It is well known that for this distribution, the conditional distribution of X1

given X2 � x2 and the conditional distribution of X2 given X1 � x1 are exponential+Also, the marginal distributions of X1 and X2 are exponentials+ However, the con-ditional distribution of X1 given X2 � x2 and that of X2 given X1 � x1 are not expo-nential+ For other properties of this model, see Arnold @4# +


It can be easily verified that in this case

g~ y16Y2 � y2 !� ~a � cy2 !e�ay1�cy1 y2, y1 � 0, y2 � 0+

This shows that the distribution of Y1 given Y2 � y2 is exponential with parametera � cy2+ Similarly, the distribution of Y2 given Y1 � y1 is exponential with param-eter b � cy1+

We will now obtain the equilibrium distributions in the case of series and par-allel systems+

Series System: In this case, we observe T1 � Min~X1, X2! and the survivalfunction is given by

OF~t ! � exp~�at � bt � ct 2 !, t � 0,

and

E~T1! �1

2c�102e ~~a�b!204c!G�1

2,~a � b!2

4c�,

where

G~m, x! ��x

`

t m�1e�t dt;

see Gupta, Tajdari, and Bresinsky @20# + The above two expressions yield thep+d+f+ of the equilibrium distribution in the case of a series system+

Parallel System: In this case, we observe T2 � Max~X1, X2! and the survivalfunction is given by

OF~t ! � e�at � e�bt � e�at�bt�ct 2

, t � 0,

and

E~T2 ! �a � b

ab� E~T1!+

The above two expressions yield the equilibrium distribution in the case of aparallel system+

Example 6.2: Generalized bivariate Pareto distribution with survival function

OF~x1, x2 ! �1

@1 � a~x1 � x2 !� a2bx1 x2 #c,

x1, x2 � 0, a, c � 0, 0 � b � 1 � c+

330 R. C. Gupta

The above survival function was obtained as a mixture of independent exponentialrandom variables; see Hutchinson and Lee @24, p+ 118# +

It can be verified that, in this case,

g~ y16Y2 � y2 ! �OF~ y1, y2 !

P~X2 � y2 !E~X16X2 � y2 !

� � 1 � ay2

1 � a~ y1 � y2 !� a2by1 y2c 1

E~X16X2 � y2 !,

where

E~X16X2 � y2 ! ��0

`� 1 � ay2

1 � a~ y1 � y2 !� a2by1 y2c

dy1

� ~c � 1!~a � a2by2 !~1 � ay2 !

c�1

@1 � a~ y1 � y2 !� a2by1 y2 #c+

We will now obtain the equilibrium distributions in the case of series and par-ralel systems+

Series System: In this case, we observe T1 � Min~X1, X2! and the survivalfunction is given by

OF~t ! �1

~1 � 2at � a2bt 2 !c, t � 0,

and

E~T1! �2c�102

aMb�1

b� 1�~1�2c!04

G�c �1

2�b~2c � 1,1!Pc�302102�c� 1

Mb�,

c �1

2,

where

Pnµ~z! �

1

G~1 � µ!� z � 1

z � 1�µ02

2 F1��n,n� 1,1 � µ,1 � z

2�

is the associated Legendre function of the first kind; see Gupta et al+ @20# +The above two expressions yield the p+d+f+ of the equilibrium distribution in

the case of a series system+


Parallel System: In this case, we observe T2 � Max~X1, X2! with survivalfunction

OF~t ! �2

~1 � ab!c�

1

~1 � 2at � a2bt 2 !c

and

E~T2 ! �2

a~c � 1!� E~T1!+

The above two expressions yield the distribution of the equilibrium distribu-tion in the case of a parallel system+

7. SOME CONCLUSIONS AND COMMENTS

In this article we have presented the equilibrium distribution including its higherderivates from a reliability point of view+ Since the equilibrium distribution is inti-mately connected with the original distribution, it provides a useful tool in studyingseveral important properties of the original distribution+ Aging properties of equi-librium distributions and their stochastic orderings with the original distributionsare explored+ The relation between the equilibrium distribution of a series systemand a series system of equilibrium distributions, consisting of two components, isinvestigated+ Extension to bivariate equilibrium distributions is provided along withsome examples+ It is hoped that this article will be useful to theoreticians and prac-titioners in studying the applications of equilibrium distribution in reliability+

AcknowledgmentThe author is thankful to the Editor Sheldon Ross for some useful comments that enhanced thepresentation+

References

1+ Abouammoh, A+M+ ~1988!+ On the criteria of the mean remaining life+ Statistics and ProbabilityLetters 6: 205–211+

2+ Abouammoh, A+M+ & Ahmed, A+N+ ~1988!+ The new better than used failure rate class of life dis-tributions+ Advances in Applied Probability 20: 237–240+

3+ Abouammoh, A+M+, Kanjo, A+, & Khalique, A+ ~1990!+ On aspects of variance remaining life dis-tributions+ Microelectron Reliability 30~4!: 751–776+

4+ Arnold, B+C+ ~1995!+ Conditional survival models+ In N+ Balakrisnan ~ed+!, Recent advances in lifetesting and reliability+ Boca Raton, FL: CRC Press, pp+ 589– 601+

5+ Basu,A+P+ & Ebrahimi, N+ ~1984!+ On k-order harmonic new better than used in expectation distri-bution+ Annals of the Institute of Statistical Mathematics A 36: 87–100+

6+ Basu, A+P+ & Ebrahimi, N+ ~1985!+ Corrections to “on k-order harmonic new better than used inexpectation distributions+” Annals of the Institute of Statistical Mathematics A 37: 365–366+

7+ Belzunce, F+, Candel, J+, & Ruiz, J+M+ ~1996!+ Dispersive orderings and characterizations of ageingclasses+ Statistics and Probability Letters 28: 321–327+

332 R. C. Gupta

8+ Bhattacharjee, M+C+ & Sethuraman, J+ ~1990!+ Families of life distributions characterized by twomoments+ Journal of Applied Probability 27: 720–725+

9+ Bon, J-L+ & Illayk,A+ ~2005!+Ageing properties and series systems+ Journal of Applied Probability42: 279–286+

10+ Brown, M+ ~2006!+ Exploiting the waiting time paradox+ Probability in the Engineering and Infor-mational Sciences 20: 195–230+

11+ Bryson,M+C+ & Siddiqi,M+M+ ~1969!+ Some criteria for aging+ Journal of the American StatisticalAssociation 64: 1472–1483+

12+ Deshpande, J+V+, Kochar, S+C+, & Singh, H+ ~1986!+ Aspects in positive ageing+ Journal of AppliedProbability 23: 74–78+

13+ Deshpande, J+V+, Singh, H+, Bagai, I+, & Jain, K+ ~1990!+ Some partial orderings describing positiveageing+ Communications in Statistics: Stochastic Models 6~3!: 471– 481+

14+ Fagiuoli, E+& Pellerey, F+ ~1993!+ New partial orderings and applications+ Naval Research LogisticsQuarterly 40: 829–842+

15+ Gupta, P+L+ & Gupta, R+C+ ~1983!+ On the moments of residual life in reliability and some charac-terization results+ Communications in Statistics: Theory and Methods 12: 449– 461+

16+ Gupta, R+C+ ~1987!+ On the monotonic properties of residual variance and their applications in reli-ability+ Journal of Statistical Planning and Inference 16: 329–335+

17+ Gupta, R+C+ & Kirmani, S+N+U+A+ ~1998!+ Residual life function in reliability studies+ In A+P+ Basu,S+K+ Basu, & S+ Mukhopadhyay ~eds+!, Frontiers in reliability+ River Edge, NJ:World Scientific+

18+ Gupta, R+C+ & Kirmani, S+N+U+A+ ~2000!+ Residual coefficient of variation and some characteriza-tion results+ Journal of Statistical Planning and Inference 91: 23–31+

19+ Gupta, R+C+, Kirmani, S+N+U+A+, & Launer, R+L+ ~1987!+ On life distributions having monotoneresidual variance+ Probability in the Engineering and Informational Sciences 1: 299–307+

20+ Gupta, R+C+, Tajdari, M+, & Bresinsky, H+ ~2006!+ Some general results for moments in bivariatedistributions+ Submitted for publication+

21+ Gupta, R+P+ & Sankaran, P+G+ ~1998!+ Bivariate equilibrium distribution and its applications to reli-ability+ Communications in Statistics: Theory and Methods 27~2!: 385–394+

22+ Hesselager, O+ ~1995!+ Order relations for some distributions+ Insurance: Mathematics and Eco-nomics 16: 129–134+

23+ Hesselager, O+, Wang, S+, & Willmot, G+ ~1997!+ Exponential and scale mixtures and equilibriumdistributions+ Scandinavian Acturial Journal 2: 125–142+

24+ Hutchinson, T+P+ & Lee, C+D+ ~1991!+ The engineering statistician’s guide to continuous bivariatedistributions+ Adelaide, Australia: Rumsby Scientific Publishing+

25+ Klar, B+ & Muller,A+ ~2003!+ Characterizations of classes of lifetime distributions generalizing theNBUE class+ Journal of Applied Probability 40: 20–32+

26+ Klefsjo, B+ ~1982!+ The HNBUE and HNWUE classes of distributions+ Naval Research LogisticsQuarterly 29: 331–344+

27+ Klefsjo, B+ ~1983!+A useful ageing property based on Laplace transform+ Journal of Applied Prob-ability 20: 615– 626+

28+ Launer, R+L+ ~1984!+ Inequalities for NBUE and NWUE life distributions+ Operations Research 32:660– 667+

29+ Li, H+ & Zhu, H+ ~1994!+ Stochastic equivalence of ordered random variables with applications inreliability theory+ Statistics and Probability Letters 20: 383–393+

30+ Loh, W+Y+ ~1984!+ A new generalization of the class of NBU distributions+ IEEE Transactions onReliability 33: 419– 422+

31+ Navarro, J+, Franco,M+, & Ruiz, J+M+ ~1998!+ Characterization through moments of the residual lifeand conditional spacings+ Sankhya, Series A 60: 36– 48+

32+ Shaked, M+ ~1981!+ Extensions of IHR and IHRA ageing notions+ SIAM Journal of Applied Math-ematics 40~3!: 542–550+

33+ Shaked,M+ & Shanthikumar, J+G+ ~1994!+ Stochastic orders and their applications+ New York:Aca-demic Press+


34+ Singh, H+ ~1989!+ On partial ordering of life distributions+ Naval Research Logistics Quarterly 36:103–110+

35+ Singh, H+ & Deshpande, J+V+ ~1985!+ On some new ageing properties+ Scandinavian Journal ofStatistics 12: 213–220+

36+ Stein, W+E+ & Dattero, R+ ~1999!+ Bondesson’s functions in reliability theory+ Applied StochasticModels in Business and Industry 15: 103–109+

37+ Willmot, G+E+ ~1997!+ Bounds for compound distributions based on mean residual lifetimes andequilibrium distributions+ Insurance: Mathematics and Economics 21: 25– 42+

38+ Willmot, G+E+, Drekic, S+, & Cai, J+ ~2005!+ Equilibrium compound distributions and stop-lossmoments+ Scandinavian Acturial Journal 1: 6–24+

39+ Zarek,M+ ~1995!+ Some generalized variability orderings among life distributions with applicationsto Weibull’s and gamma distribution+ Fasciculi Mathematics 25: 197–209+

334 R. C. Gupta

role of equilibrium distribution in reliability studies

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