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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.

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

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

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

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

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

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.

The random variable

HR order (denoted as

reversed hazard (RH) order (denoted as

which denotes the reversed hazard (RH) rate order,

MRL order (denoted as

The nonnegative random variable

The nonnegative random variable

Note that, in view of a result in Lariviere [

A nonnegative measurable function

A nonnegative function

Below, we present the definition of the proportional hazard rate (PHR) order and its related proportional aging class.

Let

Consider the situation wherein

Let

Note that

The first results of this section provide an equivalent condition for the PMRL order.

The following assertions are equivalent:

First, we prove that (i) and (ii) are equivalent. Note that the MRL of

In the context of reliability engineering and survival analysis, weighted distributions are of tremendous practical importance (cf. Jain et al. [

First, we consider the following useful lemma which is straightforward and hence the proof is omitted.

Let

Let

Let

On the other hand, in many reliability engineering problems, it is interesting to study

Let

To prove the “if” part, let

Let

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.

If

First note that

Statisticians and reliability analysts have shown a growing interest in modeling survival data using classifications of life distributions by means of various stochastic orders. These categories are useful for modeling situations, maintenance, inventory theory, and biometry. In this section, we propose a new class of life distributions which is related to the MRL function. We study some characterizations, preservations, and applications of this new class. Some examples of interest in the context of reliability engineering and survival analysis are also presented.

The lifetime variable

It is simply derived that

The lifetime random variable

The lifetime random variable

Denote

The result of Theorem

If

Let

The following counterexample shows that the MRL order does not generally imply the ASMRL order and hence the sufficient condition in Theorem

Let

As an obvious conclusion of Theorem

A lifetime random variable

To prove the “if” part, note that

The following result presents a sufficient condition for a probability distribution to be ASMRL.

Let the lifetime random variable

Recall that

Note that the ratio given in (

To demonstrate the usefulness of the ASMRL class in reliability engineering problems, we consider the following examples.

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

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

Let

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

In the following, we make conditions on the random number of shocks under which

A discrete distribution

If

We may note that, for all

Let

Reliability engineers often need to work with systems having elements connected in series. Let

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

Consider

In the following we state the preservation property of the ASMRL class under weighted distribution. Let

Let

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:

closure properties of the PMRL order and the ASMRL class under convolution and coherent structures,

discrete version of the PMRL order and enhancing the obtained results related to the D-ASMRL class,

testing exponentiality against the ASMRL class.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank two reviewers for their valuable comments and suggestions, which were helpful in improving the paper. The authors would also like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this Research Group (no. RG-1435-036).