Increasing Mean Inactivity Time Ordering: A Quantile Approach

Kayid, M.; Izadkhah, S.; Alfifi, A.

doi:https://doi.org/10.1155/2018/8756969

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2018 | Article ID 8756969 | https://doi.org/10.1155/2018/8756969

Increasing Mean Inactivity Time Ordering: A Quantile Approach

M. Kayid,^1,2S. Izadkhah,³and A. Alfifi¹

Academic Editor: Alessandro Lo Schiavo

Received18 Sept 2017

Revised03 Mar 2018

Accepted25 Mar 2018

Published26 Apr 2018

Abstract

We study further the quantile mean inactivity time order. Relations between the proposed stochastic order and the other transform stochastic orders are obtained. Besides, sufficient conditions for the stochastic order are provided. Then, preservation of the order under monotone transformations, series, and parallel systems and mixtures of a general family of semiparametric distributions is studied. Examples are also given to illustrate the results.

1. Introduction

Comparisons of random variables according to stochastic orders have played a central role in reliability theory, risk theory, and other fields. There are many stochastic orders proposed in the past years giving rise to a large body of literature (cf. Shaked and Shanthikumar [1], Müller and Stoyan [2], and Belzunce et al. [3]). In order to compare the aging properties of two arbitrary life distributions, several stochastic orders, known as transform orders, providing new relationships among several popular aging notions, have been introduced (see, e.g., Nair et al. [4] and Nanda et al. (2016) and the references therein). Consider two continuous random variables and with distribution functions and and quantile functions and , respectively, for any value Denote by and the support of the random variables and , respectively, which are assumed to be intervals. One of the strongest transform stochastic orders is the convex transform order. Van Zwet [5] proposed a skewness order, called the convex transform order, which captures the property of one distribution being more skewed than the other. It is said, according to their work, that is smaller than of the convex transform order (denoted by ) when

For more properties of the convex transform order in reliability and actuarial studies we refer the readers to Barlow and Proschan (1981), Marshall and Olkin [6], Shaked and Shanthikumar [1], Kochar and Xu [7], and Barmalzan and Payandeh Najafabadi [8] among others. In terms of aging notions of lifetime distributions (that have as the common left endpoint of their supports) Kochar and Wiens [9] called the order “” the more increasing failure rate (IFR) order which is equivalent to where are the failure rates of and , respectively, provided that and are absolutely continuous with associated density functions and and survival functions and . In the literature, several weaker transform orders have also been proposed to compare the relative aging properties. Kochar and Wiens [9] proposed another stochastic order, for describing the aging phenomenon, called decreasing mean residual life order. We say is smaller than of the decreasing mean residual life order (denoted by ) whenever in which are respective mean residual life (MRL) functions of and (cf. Lai and Xie [10] for reliability properties of the MRL functions). Kochar and Wiens [9] showed that if , where , then For further properties of the order “” we refer the readers to Kochar and Wiens [9], Kochar [11], Shaked and Shanthikumar [1], and Kang and Yan [12]. Another weaker stochastic order is the star order. We say is smaller than of the star order (denoted by ) whenever From (4.B.3) in Shaked and Shanthikumar [1],

One can see Bartoszewicz [13], Li and Xu (2004), Boland et al. [14], Bartoszewicz and Skolimowska [15], Bartoszewicz and Skolimowska [16], and Kochar and Xu [17] to find further properties of the star order in the context of reliability theory. In the context of transform orders, Belzunce et al. [18] introduced a new criterion to compare risks based on the notion of expected proportional shortfall which is useful for comparing risks of different nature free of the base currency. The aim of the current investigation is to develop the study of another transform order closely related to the convex transform and the star orders, proposed by Arriaza et al. [19]. This stochastic order is similar to the order “” but considers mean inactivity times at quantiles instead of the quantile mean residual lives of the units.

2. Main Results

In this section, we have brought our main achievements. We first recall the stochastic order and its relationships with some other well-known stochastic orders. Then preservation of the order under monotone transformations, series systems, parallel systems, and mixtures of a typical family of semiparametric distributions is investigated in detail. Some examples are also included to enhance the study of the results of this section. For a nonnegative random variable with distribution function , the mean inactivity time (MIT) of is defined as (cf. Kayid and Ahmad [20]) and similarly the MIT of having distribution is given by

To relate the MIT of two lifetime units with their ages, the MITs could be evaluated at the quantiles of the underlying distributions. Given that the failure of the unit A has occurred before or at a time point , at which and the failure of unit B has taken place before or at a time point , at which , the MIT functions of random lifetime of the unit A and random lifetime of the unit B are reduced to respectively. According to Nair et al. [4], for each , and are called quantile MITs of and There is a stochastic order in the literature called location-independent riskier order that has been introduced by Jewitt [21] to compare random assets in risk analysis, which is equivalent to comparison of quantile MIT functions. Conventionally, is said to be less than in the location-independent riskier order (denoted by ) if It is trivial to see that this is equivalent to having , for all We are now ready to establish the comparison of lifetime random variables according to the ratio of their mean inactivity times at quantiles and then present our main results about the stochastic order. To be in agreement with the name of the dual order, that is, the decreasing mean residual life order, we call the quantile mean inactivity time order introduced by Arriaza et al. [19] the increasing mean inactivity time order. We bring some useful lemmas that will be used throughout this section.

Definition 1. Suppose and are two nonnegative random variables having mean inactivity time functions and , respectively. It is said that quantile mean inactivity time of is more increasing than quantile mean inactivity time of (written as ) wheneveror equivalently if

The following lemmas will be useful to derive some of our results.

Lemma 2 (Barlow and Proschan (1975, p. 120)). Let be a measure on the interval , not necessarily nonnegative. Let be a nonnegative function defined on (i)If , for all , and if is increasing, then (ii)If , for all , and if is decreasing, then

Lemma 3 (basic composition formula, Karlin [22, p. 17]). Let be a function in and let be a function in . Then the function given by is in , where , , and are real subsets of and is a -finite measure.

2.1. Relation with Other Stochastic Orders

Let and be two random variables with respective absolutely continuous distribution functions and which are assumed to be strictly increasing. Then when and Let have density function , Then it is possible to verify that Now, one can see that yields which is in and also it is evident that is in Thus, an application of Lemma 3 leads to . For the case where and do not have absolutely continuous distributions we demonstrate the same result as follows.

Theorem 4 (Arriaza et al. [19]). Let and be two continuous nonnegative random variables. Then

From (4.B.5) in Shaked and Shanthikumar [1], if, and only if, increases in Thus, as stated in Theorem 4 this is a sufficient condition for the increasing mean inactivity time order. In the following result we provide some other sufficient conditions for the order “” such that the order “” does not hold.

Theorem 5. Let and be two absolutely continuous nonnegative random variables having interval supports and finite means which have strictly increasing distribution functions. If(i)there exists such that increases in ,(ii) strictly decreases in ,(iii) increases in , then

Proof. First, we consider two arbitrary values and such that The assumption given in (i) implies that and, therefore, Now, consider Assumption (ii) provides that and further that which holds if, and only if, which is nonnegative from (iii). That is, we proved that which means that Assertions (i) and (ii) ensure that is not increasing in , for all values in Hence, and the proof is obtained.

The sufficient conditions of Theorem 5 are in the spirit of some previous results established by Belzunce et al. [23] and Belzunce and Martínez-Riquelme [24]. The next example applies Theorem 5.

Example 6. Consider two nonnegative random variables and with respective quantile functions and for Since , thus there exists the point such that is increasing in and it is strictly decreasing in That is, conditions (i) and (ii) in Theorem 5 hold true. It is also possible to see that and and that is increasing on Assumption (iii) in Theorem 5 is therefore satisfied. Hence, and .

Theorem 7. Let and be two nonnegative random variables with continuous distribution functions and , respectively.(i)If and are absolutely continuously associated with density functions and , respectively, and if , then (ii)If and have finite means such that , then

Proof. (i) From (14), implies that which by assumption yields , for all . That is, (ii) Again, by using (14) we can concludetogether with assumption giving , for all ; that is,

Corollary 8. Let and be two nonnegative continuous random variables with finite support. If then implies

Theorem 9 (Arriaza et al. [19]). Let and be two nonnegative random variables with absolutely continuous distributions. Then

Remark 10. For any , denote , where Now, implies that

2.2. Monotone Transformations

Preservation of the increasing mean inactivity time order under increasing concave transformations is obtained as follows.

Theorem 11. Let and be two nonnegative random variables with absolutely continuous distributions. If(i) is a nonnegative differentiable, strictly increasing, and concave function,(ii) is increasing in , then

Proof. First, denote by , , , and the distribution and density functions of and , respectively, which are given byTherefore, for any , we have holds if, and only if, which holds if, and only if, From (ii), for all , the following holds: Therefore, it suffices to prove thatFor two arbitrary values and in , consider the measure is defined as in the proof of Theorem 9. From (32), the assumption that implies , for all As is implied by assumption (ii), is nonnegative and decreasing in Hence Lemma 2 (ii) immediately gives which makes (39) a valid statement. This ends the proof.

2.3. Series and Parallel Systems

We consider another reliability application of the imit order. Suppose that and denote two parallel systems and that and denote two series systems each consisting of i.i.d. components. Further, assume that are i.i.d. lifetime random variables with distribution and that are i.i.d. lifetime random variables with distribution . Denote by the lifetimes of and , respectively. Further, denote the lifetimes of and , respectively. Next we focus on relations between the imit ordering of two systems lifetimes and the imit ordering of their components lifetimes.

Theorem 12. Let and be two nonnegative random variables with absolutely continuous distribution functions. Let and be i.i.d. copies of and , respectively. Then

Proof. First, denote by and the distribution functions of and , respectively, given by from which we get and, similarly, Therefore, if we denote by and the density functions of and , respectively, then for any we have Now, we can write if, and only if, or, equivalently, ifBy making the change of variable and also taking in (49), we conclude that holds if, and only if, , if, and only if,where From (50), we know that if, and only if, in which is defined as before in Theorem 9. On the other side, we obtain by assumption, as in the proof of Theorem 9, that Since is nonnegative and decreasing for any , thus an application of Lemma 2 (ii) leads to (52). Hence, the proof is completed.

Theorem 13. In the setting of Theorem 12,

Proof. Suppose that and denote the distribution functions of and , respectively, which are given by leading to By considering and as the density functions of and , respectively, for any , the following holds:Now, let us denote , for any , such that Using the identities (56) and (57), it follows, by similar arguments to those provided for the proof of Theorem 12, thatwhich is equivalent to Thus,In addition, if then is increasing in In view of (59), for all , which means thatCombining (61) and (63), we have By Lemma 2 (ii), it is implied that which is equivalent to saying that

2.4. Comparisons of Mixtures of a Family of Semiparametric Distributions

In this subsection, preservation of the order “” under mixtures of a typical family of semiparametric distributions which includes some well-known models in reliability and survival analysis is established and vice versa. Some examples of interest are given to authenticate the results. Semiparametric distributions that are distinguished by having a parameter that is itself a distribution function and thereby extending the family from which this distribution came play an important role in statistical literature (cf. Powell [25] and Marshall and Olkin [6]). In this work, we consider a typical family of semiparametric distributions that includes some well-known models such as proportional hazards and proportional reversed hazards families. Suppose that is a random variable with distribution function , and let be a parameter with parameter space , where is an arbitrary subset of (countable or uncountable). We focus on a general semiparametric family with the underlying distribution that provides a way to add a new parameter through the relationwhere being a nonnegative one to one function satisfying the following conditions:(i), for all and (ii), for all (iii), for all (iv) is strictly increasing and right continuous for all

Under conditions (i)–(iv), in (66) is a distribution function for every . By choosing a function one obtains a general form for the function in (66) asBelow we provide some choices for the function in (69) leading to several important models.(i)Order statistics: with .(ii)Lower records: , with (iii)Upper records: , with (iv)Proportional hazards: , where (v)Proportional reversed hazards: , where (vi)Upper tail distribution: with (vii)Lower tail distribution: with

In many practical situations the parameter may not be constant due to various reasons, and the contingency of heterogeneity is sometimes unpredictable and unexplained. The heterogeneity may often not be possible to be neglected. Further, it mostly happens that data from several populations is mixed and information about which subpopulation gave rise to individual data points is unavailable. There are numerous cases in practical situations in which data are coming from various sources and the statistician, therefore, needs to be aware of the initial source from which data have been derived. The mixture of the families of distributions according to a proper mixing rule is useful to model such data sets in frail populations. In the continuing part of the paper, the mixture of the family of semiparametric distributions in (66) is considered. Formally, let be a random variable (discrete or continuous) with support in having distribution function Let and be two nonnegative random variables with distributions and , respectively. Then, we shall denote by and two random variable with distributionsrespectively. Before stating the main result of this subsection, we introduce some notations. For a given function satisfying the conditions (i)–(iv) as before, setwhich is nonnegative, strictly increasing, and right continuous. Note that we could write for In view of (70) and (71), we have

The following result, under some appropriate assumptions, translates the imit order in and to the imit order between and and vice versa.

Theorem 14. Let and be two nonnegative random variables with absolutely continuous distribution functions and , and let and have distributions and as in (70) and (71), respectively.(i)If is decreasing in then (ii)If is increasing in then

Proof. First denote by and the density functions of and , respectively, and denote by the right continuous inverse function of in (72) which is given by . Appealing to the identities in (73), for any , we have and, similarly, Therefore, if, and only if, or, equivalently, ifSince thus by making a proper change of variable, one observes from (79) that is equivalent towhich holds if, and only if, in which is defined as in the proof of Theorem 9, for which (32) holds provided that By the assumption that is decreasing we can use Lemma 2 (ii) to conclude (81). This ends the proof of (i). Now, assume that and denote with From (80), we see that Besides, if then It thus follows using (80) that from which we getTherefore, we conclude from (83) and (86) that From assumption is decreasing in Now, using Lemma 2 (ii), it is deduced that That is, The proof of (ii) is complete.

Example 15. Let and each be a sequence of independent and identically distributed random variables with distribution functions and , respectively. Let be a positive integer-valued random variable, independent of ’s and of ’s with probability mass function , in which is the set of natural numbers. Denote by the extreme minimum order statistics of the two sequences and denote their distribution functions by and , respectively. It can be easily shown that where Now, since for any and , we have which is decreasing in , for all , thus will also be decreasing in where as in (72). Now, Theorem 14 (i) is applicable and provides In a similar manner, if we denote by the extreme maximum order statistics of the two sequences then they have respective distribution functions where Because is increasing in , for all , will be increasing in . Hence, by Theorem 14 (ii),

Example 16. Marshall and Olkin [26], as a general method of introducing a parameter (tilt parameter) to give more flexibility in modelling, discussed new semiparametric families of distributions. Given a distribution function , they supposed thatwhere is a positive parameter. By making a comparison between (66) and (98) we see that must be chosen as Recently, Nanda and Das [27] considered a mixture form of the distribution (98) where is taken as a random variable in order to investigate closure properties of the model with respect to some stochastic orders. For two given random variables and with respective distribution functions and , we assume that and are random variables with distributions respectively, where is a nonnegative random variable. It is possible to observe , when , is decreasing in , and in parallel, when , is increasing in Therefore, by Theorem 14 (i), if , then implies and on the other hand when , using Theorem 14 (ii), gives

Conflicts of Interest

All of the authors declare that there are no conflicts of interest related to this paper.

Acknowledgments

The first and the third authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this research, Group no. RGP-1435-036.

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Copyright © 2018 M. Kayid et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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