Increasing Mean Inactivity Time Ordering : A Quantile Approach

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.


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  −1 () = inf{ | () ≥ } and  −1 () = inf{ | () ≥ }, respectively, for any value  ∈ (0, 1).Denote by supp() and supp() 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  −1 ( ()) is convex in  ∈ supp () . ( 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 0 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 is increasing in  ∈ (0, 1) , where   () =  ()  () , for  :  () > 0,   () =  ()  () , for  :  () > 0 (3) are the failure rates of  and , respectively, provided that  and  are absolutely continuous with associated density functions  and  and survival functions  = 1 −  and  = 1 − .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  ≤ dmrl ) whenever   ( −1 ()) ( −1 ()) is increasing in  ∈ (0, 1) , (4) in which for  :  () > 0, for  :  () > 0 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 supp() = supp() = [0, ), where  ∈ (0, ∞], then For further properties of the order "≤ dmrl " 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 "≤ dmrl " but considers mean inactivity times at quantiles instead of the quantile mean residual lives of the units.

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  ∈ (0, 1),   () =   ( −1 ()) and   () =   ( −1 ()) 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  ≤ lir ) if It is trivial to see that this is equivalent to having   () ≤   (), for all  ∈ (0, 1).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  ≤ imit ) whenever or equivalently if The following lemmas will be useful to derive some of our results.
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, ( −1 ())/( −1 ()) increases in  ∈ (0, 1).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 "≤ imit " such that the order "≤  " does not hold.

Theorem 5. Let 𝑋 and 𝑌 be two absolutely continuous nonnegative random variables having interval supports and finite means which have strictly increasing distribution functions. If
Proof.First, we consider two arbitrary values  and  such that 0 ≤  <  <  0 .The assumption given in (i) implies that and, therefore, Now, consider  <  ∈ [ 0 , 1).Assumption (ii) provides that and further that Mathematical Problems in Engineering which holds if, and only if, which is nonnegative from (iii).That is, we proved that which means that  ≤ imit .Assertions (i) and (ii) ensure that ( −1 ())/( −1 ()) is not increasing in , for all values  in (0, 1).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.

Series and Parallel
Proof.First, denote by  1: and  1: the distribution functions of  1: and  1: , respectively, given by from which we get and, similarly, Therefore, if we denote by  1: and  1: the density functions of  1: and  1: , respectively, then for any  ∈ (0, 1) we have Now, we can write  1: ≤ imit  1: if, and only if, or, equivalently, if By making the change of variable  = 1 − (1 −   ) 1/ and also taking  = 1 − (1 −   ) where From (50), we know that  1: ≤ imit  1: 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 ∫  * 0 (, ) ≥ 0, for all ,  * ∈ (0, 1).Since is nonnegative and decreasing for any  ∈ (0, 1), thus an application of Lemma 2 (ii) leads to (52).Hence, the proof is completed.

Comparisons of Mixtures of a Family of Semiparametric
Distributions.In this subsection, preservation of the order "≤ imit " 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 relation where being a nonnegative one to one function satisfying the following conditions: (i) 0 ≤ (, ) ≤ 1, for all  ∈ [0, 1] and  ∈ .
(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) as Below we provide some choices for the function  in (69) leading to several important models.
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 Θ be a random variable (discrete or continuous) with support in  having distribution function Λ.Let  and  be two nonnegative random variables with distributions  and , respectively.Then, we shall denote by  * and  * two random variable with distributions respectively.Before stating the main result of this subsection, we introduce some notations.For a given function  satisfying the conditions (i)-(iv) as before, set The following result, under some appropriate assumptions, translates the imit order in  and  to the imit order between  * and  * and vice versa.(74) (ii) If ()/ is increasing in  ∈ (0, 1] then Proof.First denote by  * and  * the density functions of  * and  * , respectively, and denote by  −1 the right continuous inverse function of  in (72) which is given by  −1 () = inf{ ∈ (0, 1) | () ≥ }.Appealing to the identities in (73), for any  ∈ (0, 1), we have and, similarly, Therefore,  * ≤ imit  * if, and only if, or, equivalently, if Since  −1 (0) = 0 thus by making a proper change of variable, one observes from (79) that  * ≤ imit  * is equivalent to which holds if, and only if, in which (, ) is defined as in the proof of Theorem 9, for which (32) holds provided that  ≤ imit .By the assumption that ()/ is decreasing we can use Lemma 2 (ii) to conclude (81).This ends the proof of (i).Now, assume that  * ≤ imit  * and denote  * * (, ) = [ < ] * * (, ) with ) .(82) From (80), we see that is increasing in  ∈ (0, 1) .