Stochastic Comparisons of Weighted Distributions and Their Mixtures.

In this paper, various stochastic ordering properties of a parametric family of weighted distributions and the associated mixture model are developed. The effect of stochastic variation of the output random variable with respect to the parameter and/or the underlying random variable is specifically investigated. Special weighted distributions are considered to scrutinize the consistency as well as the usefulness of the results. Stochastic comparisons of coherent systems made of identical but dependent components are made and also a result for comparison of Shannon entropies of weighted distributions is developed.

1. Introduction

In the literature, weighted distributions have been exhaustively applied and put to use to model data in nature, as they provide more insights to provide more adequacy in modelling as a result of variety of sampling surveys (cf. Rao [1], Patil and Rao [2] and Patil [3]). Let X be a random variable with cumulative distribution function (cdf) F and probability density function (pdf) f and let is a non-negative function such that exists and is finite for all , where is an arbitrary subset of . Then is taken to be a random variable with weighted distribution associated with f, having pdf

Many families of statistical distributions hold at the disposal of the family of weighted distributions in (1) (see, e.g., the typical weighted distributions in Section 3.1 and Section 3.2). Suppose that the hazard rate function h corresponds to the pdf f so that where is the survival function of X. In spirit of Jain et al. [4], the hazard rate of is characterized by in which As for the reversed hazard rate of we have from Sunoj and Maya [5] where and is the reversed hazard rate function of X.

The density in (1) may be used to model data randomly drawn from population at a certain level of some quantity of interest. For example, could be a particular age for an individual, a certain time point or a given threshold with a specific amount. In many realistic circumstances it is acknowledged that the parameter may not be constant so that the occurrence of heterogeneity is sometimes incalculable and unexplained. In addition, it often takes place in practical situations where data from several populations is mixed. To model such data sets mixture models are used. For example, the measurements on the life lengths of a device may be gathered regardless of the manufacturer, or data may be gathered on humans without regard, say, to blood type. If the ignored variable has a bearing on the characteristic which is being measured, then the data follow a mixture model. To all intents and purposes, it is hard to find data that are not some kind of a mixture, because there is almost always some relevant covariate that is not observed.

The study of reliability properties of various mixture models has recently received much attention in the literature. When a mixture model is fitted to survival data, the mixing operation can change the pattern of aging for the lifetime unit under consideration in some favorite way (see, for example, Finkelstein and Esaulova [6], Alves and Dias [7], Arbel et al. [8], Cole and Bauer [9], Bordes and Chauveau [10], Li and Liu [11], Amini-Seresht and Zhang [12], Misra and Naqvi [13] and Badía and Lee [14]).

Mixture models capture heterogeneity in data by decomposing the population into latent subgroups, each of which is governed by its own subgroup-specific set of parameters. To represent a general formulation of the mixture model in the case of our study, consider the density associated with (1), where and G is the cdf of the random varaible . It is known that with playing the role of the weight function through which f is altered to This signifies that the mixture density in (4) can be thought as the density of a weighted distribution with weight function v for which . In situations where is designated by a discrete random variable, a finite mixture model is considered. To this end, the model (4) is developed as where represents the value of the probability mass function (pmf) of at for Throughout the paper, it is assumed that the output random variables following the mixture weighted distribution (4) have absolutely continuous distribution functions.

To the best of our knowledge, there has not been a work on the literature to argue different stochastic properties of the parametric weighted distributions as well as their mixtures in general to be attractive for broader audiences. There is a need for an effective study in this direction. The main objective of this paper is to initiate such a study to investigate the impact of the association of the model to a parameter on some general stochastic aspects of the model.

The rest of the paper is organized as follows. In Section 2, some useful notions of stochastic orders and some further stochastic properties are presented. In Section 3, some special applied weighted distributions are introduced. In Section 4, preservation of several ordinary as well as relative stochastic orderings is studied in Section 4.1. In Section 4.2, preservation properties of some stochastic orders in the extended mixture model of weighted distributions are secured and in the long run in Section 4.3, a link to information theory is provided.

2. Preliminaries

Assume that the random variables X and Y have distribution functions F and G, survival functions and , density functions f and g, hazard rate functions and and reversed hazard rate functions and , respectively. To compare the magnitude of random variables some notions of stochastic orders are introduced below.

The random variable X is said to be smaller than the random variable Y in the

usual stochastic order (denoted by

hazard rate order (denoted by

reversed hazard rate order (denoted by

likelihood ratio order (denoted by

relative hazard rate order (denoted by

relative reversed hazard rate order (denoted by

It is known that the following implications hold:

The notions of the totally positive of order 2 (TP2) and the reverse regular of order 2 (RR2) are defined as follows.

(Karlin [15]). A function for all

If and , then h is said to be TP. If and , then h is said to be RR. It is readily pointed out that the TP [RR] property of is equivalent to saying that is non-decreasing [non-increasing] in t whenever after making the conventions that when and if . In view of the foregoing statements and by assuming and and also and , one observes holds if, and only if, is RR as a function of , where is the common support of X and Y. In a similar manner we can establish that is equivalent to being TP in

3. Special Weighted Distributions

In this section, several special parametric weight functions are presented making the investigation of the main model of (4) more developed. First, some general formations of the weight function are considered by which many important families of weighted distributions are included. In all of the cases we assume that the weight function has a finite mean with respect to the underlying distribution.

3.1. Distribution-Free Weight Functions

Here, several weight functions which do not depend on the underlying distribution are given. Suppose that are two non-negative functions of x and that are two proper functions of so that the following weight functions satisfy the requirement that . Substituting any of these weight functions in the density (4) leads to a particular model that might be of some interest in a context.

(weighted power)

(weighted exponentiated)

(multiplicative)

(additive-multiplicative)

(weighted left-truncated)

(weighted right-truncated)

3.2. Semiparametric Models

Models where the parameters of interest are finite-dimensional and the nuisance parameters are infinite-dimensional are called semiparametric models. There are some choices for the weight function that are functional of the underlying distribution function including the parameter within. Below, we list some kinds of those choices whose associated weight function depend on the underlying distribution.

(Proportional hazards) where

(Proportional reversed hazards) where

(Proportional odds ratio) where and

(Upper records) in which .

(Lower records) in which .

(Residual life) where is the guaranteed survival time.

(Inactivity time) in which is the time of observation of failure.

(Scale) in which .

4. Stochastic Orderings

In this section, preservation properties of some stochastic orders under the formation of the weighted model in the fixed as well as the random levels of the parameter are studied.

4.1. Weighted Distribution with Specific Parameter

Here, in the same vein as Misra et al. [16] several preservation properties on likelihood ratio, hazard rate and reversed hazard rates orders can be established in the sense of the model (1). Suppose that is a random variable with pdf and cdf for , and assume that follows the weighted distribution of with weight function having pdf where and are two fixed numbers in In the next round, as will be presented, conditions for stochastic orders made of and to emulate the same type of stochastic orders between and are obtained.

The following Proposition is a direct conclusion of Theorem 3.2 in Misra et al. [16].

Let

If

If

If

Preservation properties of the stochastic orders considered in Proposition 1 have been procured for some special weighted distributions by Izadkhah et al. [17] including the models of proportional (reversed) hazard rates, upper (lower) records, right (left) truncation, moment generating and size-biased distributions. Izadkhah et al. [18] obtained sufficient conditions for preservation of reversed mean residual life order and Izadkhah et al. [19] presented some conditions under which the mean residual life order is preserved under weighting. For the sake of completeness, the preservation properties of the likelihood ratio, the hazard rate and the reversed hazard rates orders are studied for some of the parametric weighted distributions considered in Section 3.1 and Section 3.2. Suppose that and are two non-negative random variables with distribution functions and , survival functions and and density functions and , respectively.

(Weighted power distribution). Assume that in which

(Weighted exponentiated distribution). Consider the weight function such that

(Additive-Multiplicative weighted distribution). Let where

(Weighted left-truncated distribution). Let

with

(Weighted right-truncated distribution). Let in which

Some relative stochastic orders including the relative (reversed) hazard rate and relative mean residual life orders have attracted the attention of researchers in the last decade (cf. Di-Crescenzo and Longobardi [20], Kayid et al. [21], Misra and Francis [22], Misra et al. [23], Ding et al. [24], Ding and Zhang [25], Misra and Francis [26] and Misra and Francis [27]). We reminisce about the definition of these orders from Rezaei et al. [28] and Kayid et al. [21] [see, for example, Definition 1(v) and (vi)]. In the next theorem, the study of preservation of the relative hazard rate and the relative reversed hazard rate orders are initiated for a well-known class of semiparamtric distributions. For , denote by () the hazard rate (resp. the reversed hazard rate) of , where and is supposed to be valid as a weight function. Before stating the result, we introduce some notations. Let be two appropriate weight functions and set

Denote by partial derivative of with respect to x, that is, The symbol is used to denote the similar sign.

Suppose that

From (2), one has where

Thus, for all we have:

By assumption, is non-increasing in It suffices only to prove that:

The assumption yields for all which further concludes that for all Therefore, is non-positive (resp. non-negative) for all if, and only if, for all and for all it holds that which is validated by assumption. □

To present the result about the preservation of the relative reversed hazard rate order we introduce some other notation. Let us define for ,

Suppose stands for the partial derivative of with respect to x, that is

Let

In spirit of (3), we can write in which

For all one obtains

Since , thus is non-decreasing in It remains to demonstrate that is non-increasing in . It is known that implies for all which in turn yields for all For that reason, is non-positive (resp. non-negative) for all if, and only if, for all and for all : which holds by assumption. □

The weight functions considered in Theorems 1 and 2 encompass some particular cases which may be of independent interest. In that regard, the following corollary is resulted.

Let

If

If

Let

We only prove the assertion (i) as the proof of (ii) is similarly accomplished. Note that analogously as in the proof of Theorem 1, we can get

It can be seen that, for all which is non-positive if, and only if, or equivalently if is non-increasing in according which the ratio is also non-increasing in that is, The proof is complete. □

The following corollary is a useful observation in the context of Theorem 3 as it illustrates that a typical parametric family of weighted distributions enjoys the relative hazard rate and the relative reversed hazard rate ordering properties in some cases.

Suppose that the random variable For

In reliability and survival theories, feature of ordering for lifetime of coherent systems is a relevant subject to be studied. To this end, Navarro et al. [29] obtained a representation of the system reliability as a distorted function of the common component reliability such that , where h is an non-decreasing function depending on the structure of the underlying system and the survival copula of the joint distribution of the component lifetimes. In this context, they have shown that the reliability function of a coherent system with dependent identically distributed (DID) components can be written as a distorted function of the common component reliability function. The following lemma is due to Navarro et al. [29].

Let where

In the set up of the particular weighted distributions given in Section 3.2, the survival function of the arisen weighted distribution can be commuted to a distorted survival function, as specified earlier in Lemma 1, for which the domination function is characterized by the associated weight function. To this purpose, consider the weight function and notice that in this case has the survival function where plays the role of a parametric domination function. Note that is a non-decreasing continuous function with and . In the reversed direction, if is a distortion (domination) function and for any then and thus Therefore, there is a unique relationship between and that is the studies of weighted distributions in the context of semiparametric models entertain the studies of distorted survival functions and vice versa. The parameter may be an appropriate quantity that affects the magnitude of system’s lifetime. In the case when DID components construct the system, may be related to the dependency of the component lifetimes in a way that the survival copula in Lemma 1 depends on . For instance, in the case where the Archimedean copula or the FGM copula is adopted to model the association of lifetime of components in a coherent system. The following results are useful to analysis of relative ordering properties of coherent systems as to the best of our knowledge such a study has not been developed in the literature thus far. The following proposition is a direct consequence of Theorem 3.

Let

The following example illustrates an application of Proposition 2.

Suppose that denote the lifetime of two coherent systems. According to Table I in Navarro et al. [

Hence,

In the following example, we show that Proposition 2 can also be applied to systems with DID components.

Suppose that where

Thus,

The following example reveals a relative ordering property in the Marshall-Olkin family of distributions.

Suppose that the incorporated weight function is where so that

It follows that that is

4.2. Comparisons of Mixture Weighted Distribution

In this segment, the problem of preservation of a number of stochastic orderings in the mixture weighted model is investigated. The study is carried out in two different settings, where firstly the random parameter varies in distribution while the underlying distribution remains unchanged and secondly the underlying distribution is changed in the case when the random parameter is fixed in distribution. The results obtained by Kayid et al. [32] are developed to entertain more dynamic weighted distributions.

It is followed up that some stochastic orders of random parameters as well as the underlying random variables are transmitted to the random variables with the associated mixture weighted distribution. Give thought to as a random variable with the pdf , the cdf and the sf for Contemplate the random variable having pdf from which the cdf and the sf of are procured after somewhat plain algebraic calculations, respectively, by where the bivariate functions A and B and the function are all determined as earlier in Section 1. In the rest of the paper, it is taken for granted that the random variables and are independent. Denote by and the hazard rate and the reversed hazard rate of , respectively. It can be seen, after some integral calculation, that and

The following result demonstrates the likelihood ratio order preservation in the model (9).

Let

It is not impenetrable to realize that if, and only if, is TP in In spirit of (9), one gets

By the assumption of we can rely on the fact that is TP (resp. RR) in It is also obvious that is TP in The general composition theorem of Karlin [15] concludes the desired result. □

Let

We prove the non-parenthetical part. The parenthetical part of the theorem can be similarly proved. In consideration of the second identity in (10) one observes where

Take into account that such that

By assumption is TP in and is TP in Hence, the general composition theorem of Karlin [15] concludes that is TP in Since and since is TP in then is non-decreasing in thus is TP in and further it is non-decreasing in It can be readily claimed by assumption that is TP in On account of Lemma 4.2 in Li and Xu [33] we deduce that is TP in and the result follows. □

In the setup of the model (9), the reversed hazard rate order of the random parameters is relocated into the overall random variables.

Let

The non-parenthetical part is only proved since the proof for the parenthetical part is analogously carried out. In view of the former identity in (10), it is inferred that if, and only if, for all Let us explicate that and

It can be seen that so that

From assumption is TP in and is TP in The general composition theorem of Karlin [15] therefore is applied to draw the inference that is TP in For that reason, for all and

It is clearly seen that for Since is TP in thus is non-increasing in and in addition is non-decreasing in On that account, is non-negative and also non-increasing in for all and for all The proof is completed by Theorem 1.B.48 of Shaked and Shanthikumar [34]. □

The weight functions brought in Section 3.1 and Section 3.2 are all TP (or RR) at least under some (mild) condition. That is, they are applicable to develop the the and the orders from the random parameter into the mixture (average) variable in the model of (9) according to the result of Theorems 4–6, respectively. It is remarkable that

is TP in whenever is non-decreasing (resp. non-increasing) in and is non-decreasing (resp. non-increasing) in .

is TP in whenever is non-decreasing (resp. non-increasing) in and is non-decreasing (resp. non-increasing) in .

is TP in .

is TP in whenever is non-increasing (resp. non-decreasing) in and is non-decreasing (resp. non-increasing) in .

is TP in whenever is non-decreasing (resp. non-increasing) in and is non-decreasing (resp. non-increasing) in .

( is TP in whenever is non-decreasing (resp. non-increasing) in and is non-decreasing (resp. non-increasing) in .

is RR in

is TP in .

is TP in .

is TP in .

is RR in

is TP in whenever f, as the density function of X, is log-convex on and it is RR in provided that f is log-concave on

is RR in whenever f is log-convex on and it is TP in when f is log-concave on

is TP in whenever has a log-convex density function while it is RR in provided that has a log-concave density function.

In a modified setup of the mixture weighted model, we consider the case where is a lifespan with pdf and cdf for Then is taken as the random variable with average density where g stands for the pdf of and for In the new setting with the mixture model given in (13) two baseline variables and are implicated. The random lifetimes and are assumed to have density functions and respectively. The mixing variable shares an equal impact upon the construction of the mixture densities.

Let

For suppose that and represent the hazard rate functions of and respectively. In a same manner as in (12), where | for It is known that if, and only if, || for all Therefore, for all and By assumption, for all Hence,

Since is non-increasing in thus by assumption, for all By Lemma 7.1(b) of Barlow and Proschan [30] to (14) we attain the proof. □

The last result establishes the reversed hazard rate ordering preservation in the baseline-varied mixture weighted model of (13).

Let

First, we denote by and the reversed hazard rate functions of and respectively, for For all where for The order relation || for all yields for all and Thus

It can be seen that is non-decreasing in From assumption, for all Lemma 7.1(a) of Barlow and Proschan [30] is applicable in (15) and provides the proof. □

4.3. A Link to Information Theory

The concept of entropy in information theory has played a prominent role in a broad area of science including statistical thermodynamics, urban and regional planning, business, economics, finance, operations research, queueing theory, spectral analysis, image reconstruction, biology and manufacturing (see, for example, El Gamal and Kim [35], Brillouin [36], Khinchin [37] and Grant [38]).

Stochastic comparisons of distributions have found a link to information theory in the literature (see, for instance, Ebrahimi et al. [39], Belzunce et al. [40], Nanda and Prasanta [41], Qiu [42], Toomaj and Di Crescenzo [43] and Toomaj and Di Crescenzo [44]).

Here before closing the paper, we impose a stochastic ordering property that leads to ordering of entropies of weighted distributions with weight functions given in Section 3.2. The extension of the Shannon entropy from the discrete case to the absolutely continuous case when dealing with lifetime random variables is defined by where f is the pdf of non-negative random variable X with an absolutely continuous distribution function. Note that log, with convention stands for the natural logarithm.

However, it is found that the entropy is related to the concept of dispersion of (random) variables. Being aware of this certitude, it is useful to concentrate on dispersion measures of probability distributions as well as their related stochastic dispersion orderings.

Let us recall from Shaked and Shanthikumar [34] that X with the pdf f and the cdf F is less (or equal) than (with) Y with the pdf g and the cdf G in dispersive order (denoted by ) whenever

It follows from (3.B.25) in Shaked and Shanthikumar [34] that

In spirit of Theorem 3.B.20(a) and Theorem 3.B.20(b) in Shaked and Shanthikumar [34] if X or Y has an increasing hazard rate function, then and if X or Y has a decreasing hazard rate function, then

In accordance with Corollary 4.4 in Bartoszewicz [45], if X or Y has a decreasing reversed hazard rate function, then and if X or Y has an increasing reversed hazard rate function, then

If X and Y are two random variables with supports and , respectively, then according to Theorem 3.B.13(a) in Shaked and Shanthikumar [34] when and also according to Theorem 3.B.13(b) in Shaked and Shanthikumar [34] when

The weight functions and considered in the following theorem depend on x only through and respectively.

Let

Take and observe that, for all and also

Note that where is obviously independent of the underlying distribution. In a similar manner,

From (3.B.23) in Shaked and Shanthikumar [34], implies that for all Thus, we conclude that

As a result, for all we deduce □