Reliability Analysis of the Proportional Mean Residual Life Order

The concept of mean residual life plays an important role in reliability and life testing. In this paper, we introduce and study a new stochastic order called proportional mean residual life order. Several characterizations and preservation properties of the new order under some reliability operations are discussed. As a consequence, a new class of life distributions is introduced on the basis of the anti-star-shaped property of the mean residual life function. We study some reliability properties and some characterizations of this class and provide some examples of interest in reliability.


Introduction
Stochastic orders have shown that they are very useful in applied probability, statistics, reliability, operation research, economics, and other related fields.Various types of stochastic orders and associated properties have been developed rapidly over the years.Let  be a nonnegative random variable which denotes the lifetime of a system with distribution function , survival function  = 1 − , and density function .The conditional random variable   = ( −  |  > ),  ≥ 0, is known as the residual life of the system after  given that it has already survived up to .The mean residual life (MRL) function of  is the expectation of   , which is given by ()  () ,  > 0, 0,  ≤ 0. ( The MRL function is an important characteristic in various fields such as reliability engineering, survival analysis, and actuarial studies.It has been extensively studied in the literature especially for binary systems, that is, when there are only two possible states for the system as either working or failed.Another useful reliability measure is the hazard rate (HR) function of  which is given by   () =  ()  () ,  ≥ 0.
The HR function is particularly useful in determining the appropriate failure distributions utilizing qualitative information about the mechanism of failure and for describing the way in which the chance of experiencing the event changes with time.In replacement and repair strategies, although the shape of the HR function plays an important role, the MRL function is found to be more relevant than the HR function because the former summarizes the entire residual life function whereas the latter involves only the risk of instantaneous failure at some time .For an exhaustive monograph on the MRL and HR functions and their reliability analysis, we refer the readers to Ramos-Romero and Sordo-Díaz [1], Belzunce et al. [2], and Lai and Xie [3].Based on the MRL function, a well-known MRL order has been introduced and studied in the literature.Gupta and Kirmani [4] and Alzaid [5] were among the first who proposed the MRL order.Over the years, many authors have investigated reliability properties and applications of the MRL order in reliability and survival analysis (cf.Shaked and Shanthikumar [6] and 2 Mathematical Problems in Engineering Müller and Stoyan [7]).On the other hand, the proportional stochastic orders are considered in the literature to generalize some existing notions of stochastic comparisons of random variables.Proportional stochastic orders as extended versions of the existing common stochastic orders in the literature were studied by some researchers such as Ramos-Romero and Sordo-Díaz [1] and Belzunce et al. [2].Recently, Nanda et al. [8] gave an effective review of the different partial ordering results related to the MRL order and studied some reliability models in terms of the MRL function.
The purpose of this paper is to propose a new stochastic order called proportional mean residual life (PMRL) order which extends the MRL order to a more general setting.Some implications, characterization properties, and preservation results under weighted distributions of this new order including its relationships with other well-known orders are derived.In addition, two characterizations of this order based on residual life at random time and the excess lifetime in renewal processes are obtained.As a consequence, a new class of lifetime distributions, namely, anti-star-shaped mean residual life (ASMRL) class of life distribution, which is closely related to the concept of the PMRL order, is introduced and studied.A number of useful implications, characterizations, and examples for this class of life distributions are discussed along with some reliability applications.The paper is organized as follows.The precise definitions of some stochastic orders as well as some classes of life distributions which will be used in the sequel are given in Section 2. In that section, the PMRL order is introduced and studied.Several characterizations and preservation properties of this new order under some reliability operations are discussed.In addition, to illustrate the concepts, some applications in the context of reliability theory are included.In Section 3, the ASMRL class of life distributions is introduced and studied.Finally, in Section 4, we give a brief conclusion and some remarks of the current research and its future.
Throughout this paper, the term increasing is used instead of monotone nondecreasing and the term decreasing is used instead of monotone nonincreasing.Let us consider two random variables  and  having distribution functions  and , respectively, and denote by () and () their respective survival (density) functions.We also assume that all random variables under consideration are absolutely continuous and have 0 as the common left endpoint of their supports, and all expectations are implicitly assumed to be finite whenever they appear.In addition, we use the notations R = (−∞,∞), R + = (0,∞),  = denotes the equality in distribution, and  V is the weighted version of  according to the weight V.

Proportional Mean Residual Life Order
For ease of reference, before stating our main results, let us recall some stochastic orders, classes of life distributions, and dependence concepts which will be used in the sequel.Definition 1.The random variable  is said to be smaller than  in the (i) HR order (denoted as ≤ HR ) if (ii) reversed hazard (RH) order (denoted as ≤ RH ) if which denotes the reversed hazard (RH) rate order, (iii) MRL order (denoted as Definition 2 (Lai and Xie [3]).The nonnegative random variable  is said to have a decreasing mean residual life (DMRL) whenever the MRL of  is decreasing.
Definition 3 (Lariviere and Porteus [9]).The nonnegative random variable  is said to have an increasing generalized failure rate (IGFR) whenever the generalized failure rate function   of  which is given by   () =   () is increasing in  ≥ 0. Note that, in view of a result in Lariviere [10],  has IGFR property if and only if ≤ HR , for all  ∈ (0, 1], or equivalently if ≤ HR , for any  ≥ 1.
Definition 4 (Karlin [11]).A nonnegative measurable function ℎ(, ) is said to be totally positive of order 2 (TP 2 ) in  ∈ R and  ∈ R, whenever         ℎ ( Consider the situation wherein  denotes the risk that the direct insurer faces and  the corresponding reinsurance contract.One important reinsurance agreement is quotashare treaty defined as () = , for  ∈ (0, 1].The random variable  denotes the risk that an independent insurer faces.Insurers sometimes seek a quota-share treaty when they require financial support from their reinsurers, thus maintaining an adequate relation between net income and capital reserves.Motivated by this, we propose the following new stochastic order.Definition 7. Let  and  be two nonnegative random variables.The random variable  is smaller than  in the PMRL order (denoted as ≤ P-MRL ), if ≤ MRL , for all  ∈ (0, 1].
The first results of this section provide an equivalent condition for the PMRL order.Theorem 9.The following assertions are equivalent: (ii)   () ≤   (), for all  ≥ 0, and each  ∈ (0, 1]; Proof.First, we prove that (i) and (ii) are equivalent.Note that the MRL of  as a function of  is given by   (/), for all  ≥ 0 and for any  ∈ (0, 1].Now, we have ≤ P-MRL  if, for all  ∈ (0, 1], it holds that where  =   . To prove that (ii) and (iii) are equivalent, we have It is obvious that the last term is nonnegative if and only if   () ≤   (), for all  ≥ 0 and for any  ∈ (0, 1]. In the context of reliability engineering and survival analysis, weighted distributions are of tremendous practical importance (cf.Jain et al. [12], Bartoszewicz and Skolimowska [13], Misra et al. [14], Izadkhah and Kayid [15], and Kayid et al. [16]).In renewal theory the residual lifetime has a limiting distribution that is a weighted distribution with the weight function equal to the reciprocal of the HR function.Some of the well-known and important distributions in statistics and applied probability may be expressed as weighted distributions such as truncated distributions, the equilibrium renewal distribution, distributions of order statistics, and distributions arisen in proportional hazards and proportional reversed hazards models.Recently, Izadkhah et al. [17] have considered the preservation property of the MRL order under weighted distributions.Here we develop a similar preservation property for the PMRL order under weighted distributions.For two weight functions  1 and  2 , assume that   1 and   2 denote the weighted versions of the random variables of  and , respectively, with respective density functions where . Then survival functions of   1 and   2 are, respectively, given by First, we consider the following useful lemma which is straightforward and hence the proof is omitted.Lemma 10.Let  be a nonnegative absolutely continuous random variable.Then, for any weight function  1 , where V is a weight function of the form V() =  1 (/).
Theorem 11.Let  2 be an increasing function and let  2 ()/ 1 () increase in  ≥ 0, for all  ∈ (0, 1].Then Proof.Let  ∈ (0, 1] be fixed.Then, ≤ P-MRL  gives ≤ MRL .We know by assumption that  2 is increasing and the ratio is increasing in  ≥ 0, when  = /.In view of Theorem 2 in Izadkhah et al. [17], we conclude that () V ≤ MRL   2 .Because of Lemma 10 and because the equality in distribution of () V and   1 implies the equality in their MRL functions, it follows that   1 ≤ MRL   2 .So, for all  ∈ (0, 1] we have On the other hand, in many reliability engineering problems, it is interesting to study   = [ −  |  > ], the residual life of  with a random age .The residual life at random time (RLRT) represents the actual working time of the standby unit if  is regarded as the total random life of a warm standby unit with its age .For more details about RLRT we refer the readers to Yue and Cao [18], Li and Zuo [19], and Misra et al. [20], among others.Suppose that  and  are independent.Then, the survival function of   , for any  ≥ 0, is given by Theorem 12. Let  and  be two nonnegative random variables.  ≤ P-MRL  for any  which is independent of , if and only if Proof.To prove the "if " part, let   ≤ P-MRL  for all  ≥ 0. It then follows that, for all  > 0 and  ∈ (0, 1], By integrating both sides of ( 16) with respect to  through the measure , we have which is equivalent to saying that   ≤ P-MRL , for all 's that are independent of .For the "only if " part, suppose that   ≤ P-MRL  holds for any nonnegative random variable .
Then   ≤ P-MRL , for all  ≥ 0, follows by taking  as a degenerate random variable.

𝐹 (𝑡 + 𝑥 − 𝑧) 𝑑𝑀 (𝑧) . (19)
In the literature, several results have been given to characterize the stochastic orders by the excess lifetime in a renewal process.Next, we investigate the behavior of the excess lifetime of a renewal process with respect to the PMRL order.
Proof.First note that   ≤ P-MRL , for all  ≥ 0, if and only if for any  ≥ 0,  > 0, and  ∈ (0, 1] In view of the identity of ( 19) and the inequality in ( 20 Hence, it holds that, for all  ≥ 0,  > 0 and for any  ∈ (0, 1], which means ()≤ P-MRL (0) for all  ≥ 0. The following counterexample shows that the MRL order does not generally imply the ASMRL order and hence the sufficient condition in Theorem 18 cannot be removed.
As an obvious conclusion of Theorem 18 above and Theorem 2.9 in Nanda et al. [8] Proof.To prove the "if" part, note that   () =   (/), for each  ∈ (0, 1] and any  > 0. Take  = , for each  ∈ (0, 1] one at a time, as a degenerate random variable implying ≤ MRL , for all  ∈ (0, 1], which means  ∈ ASMRL.For the "only if " part, assume that  has distribution function .From the assumption and the well-known Fubini theorem, for all  > 0, it follows that That is, ≤ MRL .
The following result presents a sufficient condition for a probability distribution to be ASMRL.Theorem 20.Let the lifetime random variable  be IGFR.Then,  is ASMRL.
Proof.Recall that  is IGFR if and only if   () is increasing in  ≥ 0. Because of the identity we can write, for all  > 0, Thus,   ()/ is decreasing in  > 0 if and only if the ratio Let as a function of  = 1, 2 and of  > 0, where Note that the ratio given in (29) is increasing in  > 0 if and only if  is TP 2 in (, ) ∈ {1, 2} × (0, ∞).From the assumption, since   () is increasing, then  is TP 2 in (, ) ∈ {1, 2} × (0, ∞).Also it is easy to see that  is TP 2 in (, ) ∈ (0, ∞) × (0, ∞).By applying the general composition theorem of Karlin [11] to the equality of (30), the proof is complete.
To demonstrate the usefulness of the ASMRL class in reliability engineering problems, we consider the following examples.
Example 21.The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering.It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameter.Let  have the Weibull distribution with survival function The HR function is given by   () =    −1 .Thus we have   () = ()  which is increasing in  > 0 for all parameter values and hence according to Theorem 20 is ASMRL.
Example 22.The generalized Pareto distribution has been extensively used in reliability studies when robustness is required against heavier tailed or lighter tailed alternatives to an exponential distribution.Let  have generalized Pareto distribution with survival function ,  ≥ 0,  > 0,  > 0. (33) The HR function is given by   () = (1 + )/( + ).In the context of reliability theory, shock models are of great interest.The system is assumed to have an ability to withstand a random number of these shocks, and it is commonly assumed that the number of shocks and the interarrival times of shocks are s-independent.Let  denote the number of shocks survived by the system, and let   denote the random interarrival time between the ( − 1)th and th shocks.Then the lifetime  of the system is given by  = ∑  =1   .Therefore, shock models are particular cases of random sums.In particular, if the interarrivals are assumed to be s-independent and exponentially distributed (with common parameter ), then the distribution function of  can be written as where   = [ ≤ ] for all  ∈  (and  0 = 1).Shock models of this kind, called Poisson shock models, have been studied extensively.For more details, we refer to Fagiuoli and Pellerey [24], Shaked and Wong [25], Belzunce et al. [26], and Kayid and Izadkhah [27].
In the following, we make conditions on the random number of shocks under which  has ASMRL property.First, let us define the discrete version of the ASMRL class.Proof.We may note that, for all  > 0, Hence  is ASMRL if and only if is increasing in , or equivalently if for  ∈ {1, 2} and  ∈ R + , where By appealing to Lemma 26, it follows that   ≤ MRL   ,  = 1, 2, . . ., , for all  ∈ (0, 1].That is,   ,  = 1, 2, . . ., , is ASMRL.Hence, the ASMRL property passes from the lifetime of the series system to the lifetime of its i.i.d.components.Accelerated life models relate the lifetime distribution to the explanatory variables (stress, covariates, and regressor).This distribution can be defined by the survival, cumulative distribution, or probability density functions.Nevertheless, the sense of accelerated life models is best seen if they are formulated in terms of the hazard rate function.In the following example, we state an application of Theorem 16 in accelerated life models.

Conclusion
Due to economic consequences and safety issues, it is necessary for the industry to perform systematic studies using reliability concepts.There exist plenty of scenarios where a statistical comparison of reliability measures is required in both reliability engineering and biomedical fields.In this paper, we have proposed a new stochastic order based on the MRL function called proportional mean residual life (PMRL) order.The relationships of this new stochastic order with other well-known stochastic orders are discussed.It was shown that the PMRL order enjoys several reliability properties which provide several applications in reliability and survival analysis.We discussed several characterization and preservation properties of this new order under some reliability operations.To enhance the study, we proposed a new class of life distributions called anti-star-shaped mean residual life (ASMRL) class.Several reliability properties of the new class as well as a number of applications in the context of reliability and survival analysis are included.Our results provide new concepts and applications in reliability, statistics, and risk theory.Further properties and applications of the new stochastic order and the new proposed class can be considered in the future of this research.In particular, the following topics are interesting and still remain as open problems: (i) closure properties of the PMRL order and the ASMRL class under convolution and coherent structures, [8]inition 14.The lifetime variable  is said to have an antistar-shaped mean residual life (ASMRL), if the MRL function of  is anti-star-shaped.It is simply derived that  ∈ ASMRL whenever   ()/ is decreasing in  > 0. Useful description and motivation for the definition of the ASMRL class which is due to Nanda et al.[8]are the following.Consider a situation in which  represents the risk that the direct insurer faces and  the corresponding reinsurance contract.The ASMRL class provides that the quota-share treaty related to a risk is less than risk itself in the sense of the MRL order.In what follows, we focus on the ASMRL class as a weaker class than the DMRL class to get some basic results.First, consider the following characterization property which can be immediately obtained by Theorem 9(ii).The lifetime random variable  is ASMRL if and only if ≤ P-MRL .≤MRL  2 , for any 1 ≤  2 ∈ R + .forany  1 ≤  2 ∈ R + .(24)Bytaking=  1 / 2 and  = / 1 the above inequality is equivalent to saying that   () ≤   (), for all  ≥ 0 and for any  ∈ (0, 1].This means that  is ASMRL.The result of Theorem 16 indicates that the family of distributions   () = (/),  > 0, is stochastically increasing in  with respect to the MRL order if and only if the distribution  has an anti-star-shaped MRL function.Another conclusion of Theorem 16 is to say that ≤ P-MRL  if and only if ≤ MRL , for all  ∈ [1, ∞).Hence it holds that ≤ MRL , for all  ∈ (0, 1], which means ≤ P-MRL .The proof of the result when  is ASMRL is similar by taking the fact that  is ASMRL if and only if ≤ MRL , for all  ≥ 1, into account.Note also that ≤ P-MRL  if and only if ≤ MRL , for any  ≥ 1. (23)Proof.Denote (  ) =   , for  = 1, 2. The MRL function of (  ) is then given by  (  ) () =     (/  ), for all  ≥ 0 and  = 1, 2.In view of the fact that  1 ≤ MRL  2 , for all  1 ≤  2 ∈ R + , if and only if Theorem 18.If ≤ MRL  and if either  or  has an anti-starshaped MRL function, then ≤ P-MRL .Proof.Let ≤ MRL  and let  be ASMRL.Then, we have ≤ MRL ≤ MRL , ∀ ∈ (0, 1] .(25) , if  is DMRL, then  is ASMRL.The next result presents another characterization of the ASMRL class.Theorem 19.A lifetime random variable  is ASMRL if and only if ≤ MRL , for each random variable  with [23] we get   () = (1 + )/( + ) which is increasing in  for all parameter values and so Theorem 20 concludes that  is ASMRL.Example 23.Let   be a lifetime variable having survival function given by   () = ((  )),  ≥ 0, where   is a nonnegative random variable and  is the survival function of a lifetime variable , for each  = 1, 2. This is called scale change random effects model in Ling et al.[23].Noting the fact that ≤ MRL , for all  ≥ 1, is equivalent to saying that  is ASMRL, according to Theorem 3.10 of Ling et al.[23]if  1 ≤ RH  2 and  is ASMRL, then  1 ≥ MRL  2 .
Definition 24.A discrete distribution   is said to have discrete anti-star-shaped mean residual life (D-ASMRL) property if ∑ ∞ =   / −1 is nonincreasing in  ∈ .
Then, Theorem 19 states that, for each  with support on [0, 1], we must have   ≤ MRL   .Thus, by taking  = 1/Θ  , we must have   /Θ  ≤ MRL   .Hence, by (40) it stands that   ≤ MRL   .With a similar discussion, in a gentler environment if   ∈ ASMRL for some  = 1, 2, . . ., , then we must have   ≤ MRL   .In the following we state the preservation property of the ASMRL class under weighted distribution.Let  have density function  and survival function .The following result states the preservation of the ASMRL class under weighted distributions.The proof is quite similar to that of Theorem 11 and hence omitted.