COMMUN. STATIST.—THEORY METH., 30(11), 2447–2457 (2001)
SOME RESULTS ON REVERSED HAZARD RATE ORDERING Rameshwar D. Gupta1 and Asok K. Nanda2 1
Department of Applied Statistics and Computer Science, University of New Brunswick, 100 Tucker Park Rd., Saint John, Canada E2L 4L5 E-mail:
[email protected] 2 Department of Mathematics, Indian Institute of Technology, Kharagpur 721 302, India E-mail:
[email protected]
ABSTRACT Recently, the reversed hazard rate (RHR) function, defined as the ratio of the density to the distribution function, has become a topic of interest having applications in actuarial sciences, forensic studies and similar other fields. Here we establish results with respect to RHR ordering between the exponentiated random variables. We also address the ordering results between component redundancy and system redundancy. Both the cases of matching spares and nonmatching spares are discussed. In case of matching spares, a sufficient condition has been given for component redundancy to be superior to the system redundancy with respect to the reversed hazard rate ordering for any coherent system.
2447 Copyright & 2001 by Marcel Dekker, Inc.
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Key Words: Coherent system; Component redundancy and system redundancy; Exponentiated random variables; Matching spares
1. INTRODUCTION The concept of hazard rate is well known in the literature. This has been successfully used by Reliability Analyst, Design Engineers and Actuarists. Keilson and Sumita (1) defined an ordering, later known as reversed hazard rate ordering (Shaked and Shanthikumar, (2)), which is based on the reversed hazard rate function, defined as the ratio of the density to the distribution function as contrary to the ratio of density to the survival function in case of hazard rate function (also known as mortality function in Economics and Actuarial Science). Andersen, Borgan, Gill and Keiding (3) show that the reversed hazard rate function plays the same role in the analysis of left-censored data as the hazard rate function plays in the analysis of right-censored data. For an absolutely continuous random variable X having density function f ðtÞ and distribution function FðtÞ, the reversed hazard rate function F ðtÞ is defined as F ðtÞ ¼ f ðtÞ=FðtÞ for all t such that FðtÞ > 0. Suppose a unit (that could be human being or a system or a component of a system) has already failed by time t. Then the probability that it was functional at time t (where > 0 is small) is given approximately by :F ðtÞ. If F ðtÞ is nonincreasing in t, then the random variable X (or equivalently, its distribution function F) is said to have decreasing reversed hazard rate (DRHR). One can easily verify that X is DRHR if and only if FðxÞ is logconcave. In contrast to the logconcavity of the distribution function, logconcavity of the survival function corresponds to IHR (increasing in hazard rate) distributions. Throughout in this paper, the words increasing (decreasing) and nondecreasing (nonincreasing) are used interchangeably. If two nonnegative random variables X and Y have respective reversed hazard rate functions F ðtÞ and G ðtÞ, X is said to be larger than Y in reverrh sed hazard rate ordering (X , , Y) if F ðtÞ G ðtÞ for all t 0 (cf. Shaked and Shanthikumar (2)). It can easily be verified that the reversed hazard rate ordering is weaker than the likelihood ratio ordering yet is stronger than usual stochastic ordering. Further, neither the reversed hazard rate ordering nor the hazard rate ordering implies the other. In forensic science and in actuarial science the time elapsed since failure is a quantity of interest in order to predict the exact time of failure. Suppose the observations are made periodically and at some point of time a unit is found to be dead/failed.
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Sengupta and Nanda (4) have shown that the time since failure would be stochastically larger when the interval between scheduled observations is larger, whenever the remaining life at the beginning of the observation interval has a DRHR distribution. The reversed hazard rate order is also used in econometrics and risk theory. Consider an agent having increasing concave utility function, and a choice among several random risks. Suppose the agent maximizes his expected utility by selecting an optimal level of exposure to a specified risk. If the random risks are increasing in the reversed hazard rate order, the optimal level of exposure (that the agent chooses) decreases (Eeckhoudt and Gollier (5)). Interestingly, this does not necessarily hold when they are merely increasing according to the usual stochatic dominance of first order. Further results in this direction can be found in Kijima and Ohnishi (6). Block, Savits and Singh (7) have also discussed some interesting problems on RHR function. The organization of the paper is as follows. Section 2 includes some reversed hazard rate ordering results. Here we have noticed that if two Weibull distributions have different shape parameters, then the distributions cannot be ordered in RHR ordering and hence cannot be ordered in likelihood ratio ordering. Section 3 deals with the active redundancy results corresponding to the system versus the components. Here we give sufficient conditions under which component redundancy is better than system redundancy in reversed hazard rate ordering for any coherent system of iid (independent and identically distributed) components and iid spares. Boland and El-Neweihi (8) have shown in their Example 1 that if the distribution of the iid components does not match with that of iid spares, the component redundancy cannot be superior to system redundancy in hazard rate ordering. Here we have shown for two-component series system that the result is true for reversed hazard rate order. This result also shows that all the known results of hazard rate ordering cannot be translated to RHR ordering and hence needs separate investigation.
2. REVERSED HAZARD RATE ORDERING The Weibull family accommodates IHR and DHR distributions. But it does not allow nonmonotone hazard rate distributions which are common in practice (see Rajarshi and Rajarshi (9)). Mudholkar, Srivastava and Freimer (10) have used an extension of Weibull distributions, called exponentiated Weibull family, in reanalyzing bus-motor-failure data given in Davis (11). This distribution has one scale and two shape parameters. They have shown that the exponentiated Weibull (with three parameters)
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has a better fit compared to the two parameter Weibull or one parameter exponential, which are special cases of exponentiated Weibull. This family of distributions not only contains IHR and DHR distributions, but also nonmonotone hazard rate distributions. Further, this family of distributions is computationally convenient for censored data (Mudholkar, Srivastava and Freimer (10)). A random variable X is said to be exponentiated random variable with base distribution F if X has distribution function ½FðxÞ , for some > 0. If is an integer, X can be thought of as the lifetime of a parallel system with number of components and each component lifetime distribution FðxÞ. If is not an integer, it can be considered as the lifetime of a parallel system on n (any positive integer) components with each independent component lifetime distribution ½FðxÞ=n . Suppose X has distribution function FðxÞ and Y has distribution function ½FðxÞ . Then Y ðxÞ ¼ X ðxÞ, where X ðÞ and Y ðÞ are the reversed hazard rate functions of X and Y respectively. This model is called proportional reversed hazard model. This model has been successfully used by Sengupta, Singh and Nanda (12) to analyze two sets of data, where Cox’s Proportional Hazards model does not fit well. Gupta, Gupta and Gupta (13) discusses hazard rate properties under different situations. Let X1 and X2 be two random variables having distribution G1 and G2 , where Gi ðxÞ ¼ ½FðxÞi for i ¼ 1, 2. If the baseline distribution is the same, then there is a reversed hazard rate order between X1 and X2 , if 1 6¼ 2 . In the following result, reversed hazard rate ordering between two exponentiated random variables is given when the baseline distributions are not necessarily same. The proof is omitted. Proposition 2.1. Suppose Xi Fi ðxÞ and Yi Gi ðxÞ ¼ ½Fi ðxÞi , i ¼ 1, 2. rh rh Then Y1 Y2 if 1 2 and X1 X2 :
Looking back into the importance of the exponentiated distributions (as mentioned in the beginning of this section), below we develop some results taking particular cases of the baseline distribution. The following theorem, which gives a necessary and sufficient condition for RHR ordering between random variables having Weibull distributions, is interesting in its own right. Theorem 2.1. Let Xi have Weibull distribution with distribution function rh 1 eðt=i Þ , for i ¼ 1, 2. Then X1 X2 if and only if 1 2 ð> 0Þ, no matter what the value of ð> 0Þ is. lr
Proof: For 1 ¼ 2 , the result is trivial. Suppose, 1 > 2 . Then X1 X2 and rh hence X1 X2 .
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rh
Conversely, suppose X1 X2 . This means
1 eðt=1 Þ is increasing in t 0, 1 eðt=2 Þ which after some manipulations reduces to, for ¼ ð2 =1 Þ , t1þ ð1 Þ þ t t 0 for t 2 ð0, 1:
ð2:1Þ
Write, gð Þ ¼ t1þ ð1 Þ þ t t: One can easily verify that g0 ð Þ, the derivative of gð Þ with respect to , is zero at ¼ t ln t þ 1 t=ðt 1Þ ln t. Further more, gð0Þ ¼ gð1Þ ¼ 0. Since the function is positive at ¼ 0:5 for all t 2 ð0, 1, the function will have single mode between 0 and 1. This shows that gð Þ 0 for all 2 ½0, 1. Further, since g0 ð Þ ¼ 0 has no finite root in ð1, 1Þ, and gð2Þ ¼ tð1 tÞ2 < 0, gð Þ < 0 for all > 1. Hence, ð2:1Þ holds only for 0 1, which means 1 2 .
Corollary 2.1. Let Xi follow exponentiated exponential distribution with dis rh tribution function Fi ðtÞ ¼ 1 et=i , i ¼ 1, 2. Then X1 X2 if and only if 1 2 ð> 0Þ no matter what the value of ð> 0Þ is.
Corollary 2.2. Suppose Xi Fi ðxÞ ¼ ½1 eðx=i Þ , , > 0, i ¼ 1, 2. Then rh X1 X2 if and only if 1 2 ð> 0Þ: Remark 2.1. For an extensive study of exponentiated exponential distribution, one may refer to Gupta and Kundu (14). Following result shows, if X1 and X2 both have exponentiated exponential distribution with all parameters different, then how they will be related in the reversed hazard rate order. The proof follows from Proposition 2:1 and Theorem 2:1. i Proposition 2.2. Suppose Xi Fi ðtÞ ¼ 1 et=i , i , i > 0, i ¼ 1, 2: rh rh If 1 2 and 1 2 , then X1 X2 . Further, if 1 ¼ 2 ¼ , then X1 X2 if and only if 1 2 . Remark 2.2. Suppose Xi has exponentiated Weibull distribution given by h i i i Fi ðxÞ ¼ 1 eðx=i Þ ,
i , i , i > 0,
i ¼ 1, 2:
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If 1 6¼ 2 , then X1 and X2 cannot be reversed hazard rate ordered (and hence cannot be likelihood ratio ordered) no matter whether ’s and ’s are same or different. The different cases are shown in the following counter example. If ’s are same, but 1 6¼ 2 and 1 6¼ 2 , this boils down to one given in Proposition 2.2.
Let us write gðxÞ ¼ F1 ðxÞ=F2 ðxÞ. Below we take counter example for 1 > 2 . For 1 < 2 , the same counter examples will work. In all the cases below, we take 1 ¼ 1, 2 ¼ 0:5. Example 2.1. (a) Take 1 ¼ 2 ¼ 1, i ¼ i, i ¼ 1, 2. Then gð2Þ ¼ 1:3678794, gð2:1Þ ¼ 1:3688218, gð2:5Þ ¼ 1:363757: (b) Take, in (a) above, i ¼ 3 i, i ¼ 1, 2. Then gð3Þ ¼ 0:9438584, gð4Þ ¼ 1:1353353, gð5Þ ¼ 1:0277598: (c) Take i ¼ i ¼ 1, i ¼ 1, 2: Then gð2Þ ¼ 1:1424016, gð2:5Þ ¼ 1:1556868, gð3Þ ¼ 1:1544617: (d) i ¼ 1 and i ¼ i, i ¼ 1, 2. Then gð0:1Þ ¼ 1:2947501, gð0:5Þ ¼ 1:5311318, gð2Þ ¼ 1:509349903: (e) i ¼ 1 and i ¼ 3 i, i ¼ 1, 2. Then gð3:5Þ ¼ 1:1117171, gð4Þ ¼ 1:1145409, gð4:5Þ ¼ 1:1110961:
In the above example it is shown by taking different values of and that as long as 1 6¼ 2 , X1 and X2 cannot be ordered in the reversed hazard rate order, no matter whether 1 > ð¼ , ð¼ , 0, GðxÞ ¼ ½FðxÞ ,
x > 0:
ð2:2Þ
Then the reversed hazard rate function of G, denoted by G ðxÞ, can be written as
G ðxÞ ¼ rF ðxÞ
F ðxÞ
:
1 F ðxÞ
This shows that if F is DHR, then G is DRHR. This conclusion also follows from the fact that F and G have proportional hazard rates and hence if one is DHR, other is also DHR along with the fact that the DHR class is contained in DRHR class. The following result is easy to verify using Theorem 2:B:13 of Shaked and Shanthikumar (2).
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Theorem 2.2. For the model ð2:2Þ, (i) (ii)
lr
lr
X ðÞY if > ð ð t P ðTÞ ðSÞ , which means _ st _ T S ðTÞ ðSÞ,
ð3:3Þ
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where represents usual stochastic orders among random variables W (Shaked and Shanthikumar (2)) and the lattice operation refers to W n the usual ordering in R i.e., T S ¼ ðmaxðT1 , S1 Þ, maxðT2 , S2 Þ, . . . , maxðTn , Sn ÞÞ.
3.1. Non-matching Spares When the distribution of the lifetimes of the spares is identical with that of the original components, the spares are called matching spares; otherwise they are called non-matching spares. To be precise, if d Ti ¼ Si , i ¼ 1, 2, . . . , n, then the spares are called matching spares, otherwise they are called non-matching spares. rh st One can easily verify that X Y , ½XjX t ½YjY t for all t 0, which is again equivalent to the fact that FðtÞ=GðtÞ is increasing in t, where F and G are the distribution functions of X and Y respectively. If represents the lifetime of a parallel system, then rh
_
_
ðT SÞ ðTÞ ðSÞ W
W
is trivially satisfied, since ð T SÞ ð T Þ ð S Þ. Theorem 3.1. For a two component series system with iid components, even if the distribution of the lifetimes of the iid spares does not match with that of the components, _ rh _ T S ðTÞ ðSÞ: Proof: Let the iid component (spare) lifetimes have distribution FðGÞ and the density f ðgÞ. Then it is sufficient to show that ðtÞ ¼
2 FðtÞGðtÞ is increasing in t: ð2 FðtÞÞð2 GðtÞÞ
Taking derivative of the above expression with respect to t, we get sign
0 ðtÞ ¼ ½2 FðtÞGðtÞ½2f ðtÞ þ 2gðtÞ FðtÞgðtÞ f ðtÞGðtÞ ½4 2FðtÞ 2GðtÞ þ FðtÞGðtÞ½FðtÞgðtÞ þ f ðtÞGðtÞ,
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sign
where ‘a ¼ b’ means a and b are same in sign. This expression, after simplification reduces to 2f ðtÞð1 GðtÞÞð2 GðtÞÞ þ 2gðtÞð1 FðtÞÞð2 FðtÞÞ, which is nonnegative, implying ðtÞ is increasing in t. Hence the result is established. Remark 3.1. This result cannot be extended to likelihood ratio ordering, since, with the help of a counter example, Boland and El-Neweihi (8) have shown that the result is not true for hazard rate ordering in case of non-matching spares. This also tells us that all the results known for hazard rate order cannot be translated to the reversed hazard rate order.
3.2. Matching Spares d
In this section, we assume that Ti ¼ Si , for i ¼ 1, 2, . . . , n, i.e. the spares are matching spares. Boland and El-Neweihi (8) have shown that in case of matching spares, the redundancy at the component level is better in the hazard rate ordering than redundancy at the system level for series system. If, in addition to assuming that the lifetimes of the spares match that of the components, we allow the component lifetimes to be identically distributed, then we can obtain a sufficient condition under which the redundancy at the component level is superior to that at the system level in the reversed hazard rate ordering for any coherent system. This is given in the following theorem. Theorem 3.2. Let T1 , T2 , . . . , Tn be iid component lifetimes and S1 , S2 , . . . , Sn d be the lifetimes of the spares which are also iid. Suppose Ti ¼ Si (matching spares). Further suppose, be the lifetime of the coherent system formed from the n components and h be the system reliability function. If ð1 pÞh0 ðpÞ=1 hðpÞ is increasing in p and hðpÞ p for all rh W W p 2 ½0, 1, then ðT SÞ ðTÞ ðSÞ, where h0 ðpÞ ¼ dhðpÞ=dp. Proof: Let C ðtÞ and S ðtÞ denote the reversed hazard rate functions of the system with redundancy at the component level and that of the system with redundancy at the system level respectively. Let hkjn ðtÞ denote the reliability
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function of a k-out-of-n system. Then h0 h1j2 ð FðtÞÞ h01j2 ð FðtÞÞf ðtÞ C ðtÞ ¼ 1 h h1j2 ð FðtÞÞ 1 h1j2 ð FðtÞÞ h0 h1j2 ð FðtÞÞ h01j2 ð FðtÞÞð1 FðtÞÞ f ðtÞ : : ¼ 1 h h1j2 ð FðtÞÞ 1 FðtÞ 1 h1j2 ð FðtÞÞ
¼
ð1 FðtÞÞh0 ð FðtÞÞ h01j2 ð FðtÞÞð1 FðtÞÞ f ðtÞ : : 1 hð FðtÞÞ 1 FðtÞ 1 h1j2 ð FðtÞÞ
ð1 FðtÞÞh0 ð FðtÞÞ h01j2 ðhð FðtÞÞÞð1 hð FðtÞÞÞ f ðtÞ : : 1 hð FðtÞÞ 1 FðtÞ 1 h1j2 ðhð FðtÞÞÞ
h01j2 ðhð FðtÞÞÞh0 ð FðtÞÞf ðtÞ ¼ S ðtÞ: 1 h1j2 ðhð FðtÞÞÞ
The first inequality follows since h1j2 ðpÞ ¼ pð2 pÞ p, while the second inequality follows from the assumption that hðpÞ p.
Remark 3.2. Let us consider a series system where each component is a kj -outof-nj subsystem and at least one subsystem consists of single component. Then one can verify that (see also Boland and El-Neweihi (8) for such a system hðpÞ p. Again, ð1 pÞh0 ðpÞ=1 hðpÞ is increasing in p for any k-out-of-n system (cf. Nanda, Jain and Singh (16). So, the above result holds for such systems. In particular, this result holds for series systems of iid components and spares.
ACKNOWLEDGMENTS The work was done while the second author was visiting the University of New Brunswick, Saint John, Canada, E2L-4L5. Part of this work was supported by the grant from The Natural Sciences and Engineering Research Council of Canada, Grant No. OGP-0004850.
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3. Andersen, P. K.; Borgan, O.; Gill, R. D.; Keiding, N. Statistical Methods Based on Counting Processes, Springer Verlag, New York, 1993. 4. Sengupta, D.; Nanda, A. K. Log-Concave and Concave Distributions in Reliability. Naval Research Logistics, 1999, 46, 419–433. 5. Eeckhoudt, L.; Gollier, C. Demand for risky assets and the monotone probability ratio order. Journal of Risk and Uncertainty, 1995, 11, 113–122. 6. Kijima, M.; Ohnishi, M. Stochastic Orders and their Applications in financial optimization. Mathematical Methods of Operations Research, 1999, 50, 351–372. 7. Block, H. W.; Savits, T. H.; Singh, H. The Reversed Hazard Rate Function. Probability in the Engineering and Informational Sciences, 1998, 12, 69–90. 8. Boland, P. J.; El-Neweihi, E. Component Redundancy versus System Redundancy in the Hazard Rate Ordering. IEEE Transactions on Reliability, 1995, 44, 614–619. 9. Rajarshi, S.; Rajarshi, M. B. Bathtub Distributions: A Review. Communications in Statistics-Theory & Methods, 1988, 17, 2597–2621. 10. Mudholkar, G. S.; Srivastava, D. K.; Freimer, M. The Exponentiated Weibull Family: A Reanalysis of the Bus-Motor-Failure Data. Technometrics, 1995, 37, 436–445. 11. Davis, D. J. An Analysis of Some Failure Time Data. Journal of American Statistical Association, 1952, 47, 113–150. 12. Sengupta, D.; Singh, H.; Nanda, A. K. The Proportional Reversed Hazards Model, Technical Report, Indian Statistical Institute, Calcutta, 1999. 13. Gupta, R. C.; Gupta, P. L.; Gupta, R. D. Modeling Failure Time data by Lehmann Alternatives. Communications in Statistics—Theory & Methods, 1998, 27, 887–904. 14. Gupta, R. D.; Kundu, D. Generalized Exponential Distribution. Australian & New Zealand Journal of Statistics, 1999, 41, 173–188. 15. Barlow, R. E.; Proschan, F. Statistical Theory of Reliability and Life Testing: Probability Models, Holt, Rinehart and Winston, INC., 1975. 16. Nanda, A. K.; Jain, K.; Singh, H. Preservation of some Partial Orderings under the Formation of Coherent Systems. Statistics and Probability Letters, 1998, 39, 123–131. Received August 2000 Revised April 2001
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