Residual Probability Function, Associated Orderings, and Related Aging Classes

The concept of residual probability plays an important role in reliability and life testing. In this investigation, we study further the residual probability order and its related aging classes. Several characterizations and preservation properties of this order under some statistical and reliability operations of monotone transformation, mixture, weighted distributions, and order statistics are discussed. In addition, by comparing the original distribution with its associated equilibrium distribution with respect to the residual probability order, new aging classes of life distributions are proposed and studied. Finally, a test of exponentiality against such classes is derived and sets of real data are used as examples to elucidate the use of the proposed test for practical problems.


Introduction
The residual probability (RP) function is a well-known reliability measure which has applications in many disciplines such as reliability theory, survival analysis, and actuarial studies.Let  and  be two random lifetimes representing the lifetimes of two systems with distribution functions  and  and survival functions  = 1 −  and  = 1 − , respectively.The systems can be considered as the products of two different branches of a company.Then the quantity ( > ) gives the reliability of  relative to .In case of both  and  being distributed as Weibull, Brown and Rutemiller [1] have pointed out that to design as long-lived a product as possible one can consider the quantity ( > ) and then choose  or  when this probability is greater or less than 0.5, respectively.However, if the systems are known to have a survival age , it is important to take into account the age, when we compare the remaining lifetimes.Let   = [ −  |  > ] and   = [ −  |  > ] denote the additional residual lifetime of  and  given that the systems have survived up to age .The RP function is defined as  () =  (  >   ) ,  > 0. ( The RP function uniquely determines the distribution function of  (and hence the distribution function of ), under the condition that the ratio of the hazard rates of  and  is known.In addition, when the ratio of the hazard rates of  and  is a monotone function of time, then RP function is also a monotone function of time.As a result, the study of the properties of RP function might be important for engineers and system designers to compare the lifetime of the products and, hence, to design better products.For example, consider a series system with two independent components.If  and  denote the lifetime of the components, then clearly the lifetime of the system is  = min{, }.It is easily seen that () = ( =  |  > ), that is, the probability that the component with lifetime  causes the system failure given that the system has survived up to time  (cf.Zardasht and Asadi [2] for several reliability properties, Tan and Lü [3] for some biological background, and Lü and Chen [4], Chen et al. [5], and Zhou et al. [6] for some real world applications).
One of the main objectives of statistics is the comparison of random quantities.These comparisons are mainly based on the comparison of some measures associated with these random quantities.In many cases the researcher can express various forms of properties about the underlying distributions in terms of their survival functions, hazard rate functions, reversed hazard functions, mean residual functions, and other suitable functions of probability distributions.Comparisons of random variables based on such different functions are usually establishing partial orders among them which is known as stochastic orders.Formally, in view of the RP function, the lifetime random variable  is said to be smaller than  in the RP order (denoted by  ≤ rp ) if and only if  () ≤ 0.5, ∀ > 0. ( This stochastic order states that, between two used items of the same age  having original lifetimes  and , the one that has lifetime  has a greater chance of surviving after the time  than the item with lifetime .On the other hand, 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 (cf.Barlow and Proschan [7] and Lai and Xie [8]).The random variable  * with distribution  * () = ∫  0 ()/, for all  ≥ 0, where  is the mean of , is known in the literature as the equilibrium distribution associated with .In literature, it is found that the equilibrium distribution can be used to characterize some aging properties (cf.Bhattacharjee et al. [9], Mi [10], Bon and Illayk [11], Mugdadi and Ahmad [12], Bon and Illayk [13], Li and Xu [14], and Kayid et al. [15]).The main motivation of our work is a recent paper written by Zardasht and Asadi [2].Ordinarily, when a stochastic order is proposed in the literature, its further properties in different forms of statistical analysis become important to study.
The purpose of this paper is to achieve two goals.The first one is to provide some characterizations, preservation results, and applications for the RP order.The second goal is to propose and study a new aging notion based upon the RP comparison between a random life and its equilibrium version.The paper is organized as follows.In Section 2, some characterizations and implications regarding the RP order are provided.In that section, preservation properties under some reliability operations such as monotonic transformation and mixture are discussed.In Section 3, we investigate a new aging notion based upon the RP order and develop a nonparametric method to test exponentiality against such a strict aging property.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.It is always assumed that the nonnegative random variables  and  will have  and , respectively, as their respective density functions and   () = ()/() and   () = ()/() as their hazard rate functions.All integrals, expectations, and derivatives are implicitly assumed to exist wherever they are given.

Definitions, Characterizations, and Implications
The objective of this section is to concentrate on the relations between RP order and other well-known stochastic orders.In addition, we discuss some preservation properties of the RP order under some well-known reliability operations such as mixture, monotone transformations, weighted distributions, and order statistics.For ease of reference, before stating our main results, we present some definitions and basic properties which will be used in the sequel.For an exhaustive monograph on definitions and properties of stochastic orders we refer to Shaked and Shanthikumar [16], Di Crescenzo [17], and Alzaid and Benkherouf [18] and for aging notions we refer to Barlow and Proschan [7] and Lai and Xie [8].

Mathematical Problems in Engineering 3
The following lemma which is due to Barlow and Proschan [7] is essential in deriving our main results.Lemma 3. Let  be a real valued measure which is not necessarily nonnegative.Let ℎ be a nonnegative real valued function.If ∫ ∞  () ≥ 0 for all  ≥ 0 and if ℎ is an increasing function, then The first result shows that the RP order is stronger than the dual order.Proof.For any  ≥ 0, set from which we get, for any  ≥ 0, According to Remark 3 in Zardasht and Asadi [2], ∫ ∞  () ≥ 0, for all  ≥ 0. For a fixed  ≥ 0, take We see that ℎ is a nonnegative and increasing function in , for any  ≥ 0. So, because of Lemma 3, the nonnegativity of for any  ≥ 0. ( By taking  = lim  → ∞ ()/(), it follows that  ≥ ()/ (), for all  ≥ 0. Note that if we put  = 0 in the recent inequality we see that  ≥ 1. Hence there exists an  ≥ 1 for which () ≤ (), for all  ≥ 0. This completes the proof.
Mixture models are widely used as computational convenient representations for modeling complex probability distributions.In practical situations, it often happens that data from several populations are mixed and information about which subpopulation gave rise to individual data points is unavailable.Mixture models are used to model such data sets in nature.The next result states the preservation property of the RP order under mixture.This result strengthens the result of Theorem 8 of Zardasht and Asadi [2] to a more general setting.
Weighted distributions are useful in the context of model selection.Rao [19] introduced the family of weighted distributions in a unified way.One of the problems that has recently received much attention by many researchers is the preservation of stochastic orders by weighted distributions (cf.Bartoszewicz and Skolimowska [20], Misra et al. [21], Izadkhah and Kayid [22], and Belzunce et al. [23]).For two weight functions  1 and  2 , assume that   1 and   2 denote two random variables with respective density functions: where The next result provides a preservation property of the RP order under weighted distributions.
Proof.Because of the first stated assumption we can write, for all  ≥ 0, where , which is increasing in  ≥ 0, for all  ≥ 0, because of the second assumption, and We have which is nonnegative, for any  ≥ 0, by assumption.So, since ℎ  () is increasing in , for any fixed  ≥ 0, thus by Lemma 3 we deduce that ∫ ∞ 0 ℎ  ()() ≥ 0, for all  ≥ 0. Hence this completes the proof of the theorem.
The next example shows that the RP order is preserved under the model of proportional hazard rates (PHR).
Example 8. Suppose that  and  are two nonnegative random variables with survival functions  and , respectively, such that  ≤ rp  holds.Assume that   1 and   2 are weighted versions of  and , with weights are increasing in  ≥ 0. Thus, assumptions of Theorem 7 hold and hence The next result shows that the RP order is preserved under monotone transformation.Theorem 9. Let  be a nonnegative strictly increasing and differentiable function.Then Proof.We know that  ≤ rp  implies that Denote by   ,   ,   , and   the density and the survival functions of  and , respectively.We can see that where the change of variable  =  −1 () is imposed.Appealing to (30) the result follows.
The concept of residual life plays an important role in reliability, survival analysis, economics, and actuarial sciences.Let  be a nonnegative random variable with survival function .The random variable   = ( −  |  > ), for  : () > 0, is well known in the literature as the residual lifetime variable associated with .The random variable   represents the lifetime of a used device of age .Similarly, define   = ( −  |  > ), for  : () > 0, in which  is the survival function of .In the literature, the concept of residual life has been extended to the case where  is random.Suppose that  is a nonnegative random variable with distribution function  such that [()] > 0. Then the random variable   = ( −  |  > ) is known as the residual lifetime of  at the random time  (cf.Marshall [24], Nanda and Kundu [25], and Abouammoh et al. [26]).The following result establishes two characterizations of the RP order by means of the concepts of residual life and residual life at random time.
which means that   ≤ rp   .Conversely, if we assume that   ≤ rp   for all nonnegative random variables , independent from  and , then by taking  as degenerate random variables the result immediately follows.

Related Aging Classes
In reliability, various aging classes of life distributions have been introduced to describe several types of deterioration (improvement) that accompany aging.The past decades witnessed some aging notions based on a stochastic comparison between a random life  and its equilibrium version  * which are introduced and studied (cf.Li and Xu [14], Bhattacharjee et al. [9], and Ahmad et al. [27]).This section investigates the following new aging notion.
Definition 11.A random life  is said to be new better than renewal used in the RP order (NBRU rp ), if  * ≤ rp , or, equivalently, As the dual version, new worse than renewal used in the RP order (NWRU rp ) may be defined through  * ≥ rp .The next result states some relationships among aging classes of life distributions.
Theorem 12.We have the following assertions.
Proof.To prove (i) and (ii) note that from definition  is DMRL (IMRL) if and only if  * ≤ hr (≥ hr ).By Theorem 6 of Zardasht and Asadi [2] it follows that  * ≤ rp (≥ rp ), which means that  is NBRU rp (NWRU rp ).To prove (iii) we first need to recall from Zardasht and Asadi [2] that if  has a decreasing density function, then  ≤ rp  implies  ≤ mrl .
In addition, we know from Shaked and Shanthikumar [16] that the hazard rate order implies the mean residual life order.Now, if  is NWRU rp then  * ≥ rp , where  * has a decreasing density.Consequently,  * ≥ mrl , which is equivalent to  being IVRL.
Next we prove the preservation of NWRU rp under monotone transformations.

Theorem 13. Let 𝜙 be a nonnegative strictly increasing and convex function which is differentiable. If
Because of assumptions,   is a nonnegative increasing function which provides that, for all  ≥ 0, In addition, since   () is nonnegative and increasing in , thus by taking () =   ()[ >  −1 ()], for any fixed  ≥ 0, as a nonnegative increasing function in , in Lemma 3 we deduce that the integral given in (36) is nonnegative which validates the proof.
Before stating other results we recall the following lemma from Ahmad et al. [27].Lemma 14. Assume that () is a Lebesgue-Stieltjes measure, not necessarily positive.If ℎ 1 and ℎ 2 are two nonnegative increasing functions, and The following result states that the NBRU rp (NWRU rp ) property passes from the lifetime of a parallel system into its i.i.d.lifetime components.
Proof.We only prove the theorem for the case of NBRU rp .The other case is quite similar.Because of the identity from (36) we have that  is NBRU rp if and only if Note that where in Lemma 14 provides the proof.
Proof.We prove the theorem in the case of NBRU rp because the other case can analogously be derived.We know that   is NBRU rp if and only if By assumption, for all  ≥ 0, one obtains that where ∞  () ≥ 0, for all  ≥ 0. Because ℎ 1 =  and ℎ 2 =  are two increasing functions by assumption, thus applying Lemma 14 to (44) completes the proof.
The exponential distribution represents the lifetime of the units that never ages due to wear and tear.On the other hand, some nonparametric classes have come up in the literature testifying to how a lifetime component or/and a system ages over the time.A natural question to ask is which aging class a real data set belongs to.Thus, the problem of testing exponentiality against various nonparametric classes may be of some interest in reliability or survival analysis (cf.Lai and Xie [8]).In the rest of this section, we proceed to construct a test for exponential distribution within the NBRU rp class of life distribution.Formally, we test H:  is exponential versus K:  is NBRU rp but not exponential.The following result is substantially useful in the sequel to formalize an appropriate test statistic as a departure measure from H in favor of K.

Lemma 17.
Let  1 ,  2 ,  3 be i.i.d.copies of  which is   .Then Proof.Using the well-known Fubini theorem that  is NBRU rp implies that (48)

Mathematical Problems in Engineering
For the left hand side of (48) we have For the right hand side of the inequality in (48) we get The result now immediately follows.
By considering Δ given in Lemma 17 as a measure of departure from H in favor of K, it is noticeable that, under H, Δ = 0 and it is positive under K. Denote Given that  1 ,  2 , . . .,   is a random sample from , an unbiased estimation of Δ is which serves as an unbiased estimate of the parameter Δ = [( 1 ,  2 ,  3 )].Note that, to make Δ scale free, we only need to divide it by .Hence,  = Δ/ is a scale-free test against K, and the test statistic δ = Δ/ is asymptotically distribution-free as we show in the following result.
Proof.Using the general theory of U-statistics and von Mises statistics (see Lee [28]) as  → ∞, √( δ−) is asymptotically normal with mean 0 and variance  2 , where Thus, we have Under H,  in the above identity is exponential with mean 1.
After some calculation, the assertion follows.
In practice, we can evaluate √ δ/√43/120 and reject H if the observed value exceeds the 1 −  quantile of the standard normal distribution (0, 1).To assess the goodness, we evaluate the Pitman's asymptotic efficacy of the test: Three of the most commonly used alternatives are as follows.
The null is at  = 0 in (i) and (ii) and at  = 1 in (iii).Direct calculation for the above three alternatives gives the values 0.5708, 0.2681, and 1.4263, respectively.

Monte Carlo Null Distribution Critical Values.
In practice, simulated percentiles for small samples are commonly used by applied statisticians and reliability analysts.We have simulated the upper percentile values for 95, 98, and 99.Table 1 presents these percentile values of the statistic δ and the calculations are based on 10,000 simulated samples of sizes  = 5(5)100.

The Power of the Proposed Test.
We calculate the power of the proposed test at a significance level  using simulated number of sample 10000 for sample size  = 10, 20, and 30 and  = 1, 2, and 3 with respect to the alternatives  1 ,  2 , and  3 .Table 2 below shows the power of the test at different values of  and significance level  = 0.05.
From Table 2, it is noted that the power of the test increases when the values of the parameter  and sample size  increase, and it is clear that our test exhibits rather good powers.

Numerical Results.
To demonstrate the test method above, we apply it to the data sets.
Example 19.Consider the following data in Bhattacharjee et al. [9]; these data represent set of 40 patients suffering from Example 20.In an experiment at Florida State University to study the effect of methyl mercury poisoning on the life lengths of fish, goldfish were subjected to various dosages of methyl mercury (cf.Kochar [29]).At one dosage level the ordered times to death in week are 6.000, 6.143, 7.286, 8.714, 9.429, 9.857, 10.143, 11.571, 11.714, and 11.714.
It was found that δ = 0.01254 which is less than the critical value of Table 1.Then we accept the null hypothesis of exponentiality property.

Conclusions
The residual probability function is a new meaningful quantity in reliability theory to evaluate the lifetime of systems by taking their age into account.In this paper, we focused on two purposes.The first one was to concentrate on the residual probability order that firstly initiated by Zardasht and Asadi [2], to refine and complement some of the results given in this context.Specifically, it was obtained that the RP order directly implies the dual order which states that the survival function of the greater random variable is bounded.In addition, we considered a general preservation property of the RP order under mixture of distributions.Another result which indicates that the RP order passes from the lifetime of two series systems into the lifetime of their i.i.d.components is derived.We concluded our first goal by giving some preservation properties of the RP order under weighted distributions and increasing transformations and giving some characterization properties of the RP order via the concepts of the residual life at fixed age and the residual life at random age.The second purpose of the paper was to introduce a new aging process using the concept of residual probability order.We first gave some of its properties and then provided a test Mathematical Problems in Engineering of exponentiality against such aging notion.Some numerical results were given in order to indicate the usefulness and proficiency of the established testing procedure.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 closure properties of the PR order and the NBRU rp class under convolution and coherent structures are interesting and still remain as open problems.

Theorem 10 .
For the two lifetime distributions  and , we have (i)  ≤   if and only if   ≤    , for all  ≥ 0; (ii)  ≤   if and only if   ≤    , for all nonnegative random variables  that are independent from  and .Denote by   ,   ,   , and   the density and the survival functions of   and   , respectively.We know that   ≤ rp   , for any  ≥ 0, if and only if ∫ [2][ ()  () −  ()  ()] , (32) which is nonnegative for all  ≥ 0, and  ≥ 0, if and only if  ≤ rp .(ii)Firstassumethat≤ rp .From Zardasht and Asadi[2]we know that  ≤ rp  is equivalent to () = [  ≥   ] ≤ 0.5, for all  ≥ 0. Also, it is derived that () = [ >  > ]/[()()], for any  ≥ 0.Suppose that  has distribution .Because  is independent from  and , thus we can write

Table 1 :
The upper percentile of δ with 10000 replications.