Stochastic Comparison of Parallel Systems with LogLindley Distributed Components
Abstract
In this paper, we study stochastic comparisons of parallel systems having logLindley distributed components. These comparisons are carried out with respect to reversed hazard rate and likelihood ratio ordering.
Keywords and Phrases: Likelihood ratio order, LogLindley distribution, Majorization, Multipleoutlier model, Reversed hazard rate order, Schurconvex.
AMS 2010 Subject Classifications: 62G30, 60E15, 60K10
1 Introduction
In reliability optimization and life testing experiments, many times the tests are censored or truncated when failure of a device during the warranty period may not be counted or items may be replaced after a certain time under a replacement policy. Moreover, many reliability systems and biological organism including human life span are bounded above because of test conditions, cost or other constraints. These situations result in a data set which is modeled by distributions with finite range (i.e. with bounded support) viz. power function density, finite range density, truncated Weibull, beta, Kumaraswamy and so on (see for example, Ghitany [5], Lai and Jones [10], Lai and Mukherjee [11], Moore and Lai [17] and Mukherjee and Islam [18]).
Recently, Gmez et al. [6] introduce the logLindley (LL) distribution with parameters , written as LL(), as an alternative to the beta distribution with the probability density function given by
(1.1) 
where is the shape parameter and is the scale parameter. This distribution with a simple expression and nice reliability properties, is derived from the generalized Lindley distribution as proposed by Zakerzadeh and Dolati [21], which is again a generalization of the Lindley distribution as proposed by Lindley [14]. The LL distribution exhibits bathtub failure rates and has increasing generalized failure rate (IGFR). This distribution has useful applications in the context of inventory management, pricing and supply chain contracting problems (see, for example, Ziya et al. [22], Lariviere and Porteus [12] and Lariviere [13]), where a demand distribution is required to have the IGFR property. Moreover, it has application in the actuarial context where the cumulative distribution function (CDF) of the LL distribution is used to distort the premium principle (Gmez et al. [6]). The LL distribution is also shown to fit rates and proportions data better than the beta distribution (Gmez et al. [6]).
Order statistics play an important role in reliability optimization, life testing, operations research and many other areas. Parallel and series systems are the building blocks of many complex coherent systems in reliability theory. While the lifetime of a series system corresponds to the smallest order statistic , the same of a parallel system is represented by the largest order statistic . Although stochastic comparisons of order statistics from homogeneous populations have been studied in detail in the literature, not much work is available so far for the same from heterogeneous populations, due to its complicated nature of expressions. Such comparisons are studied with exponential, gamma, Weibull, generalized exponential or Frchet distributed components with unbounded support. One may refer to Dykstra et al. [2], Misra and Misra [16], Zhao and Balakrishnan ([23]), Torrado and Kochar [20], Kundu and Chowdhury [8], Kundu et al. [9], Gupta et al. [7] and the references there in. Moreover, not much attention has been paid so far to the stochastic comparison of two systems having finite range distributed components. The notion of majorization (Marshall et al. [5]) is also essential to the understanding of the stochastic inequalities for comparing order statistics. This concept is used in the context of optimal component allocation in parallelseries as well as in seriesparallel systems, allocation of standby in series and parallel systems, and so on, see, for instance, ElNeweihi et al. [3]. It is also used in the context of minimal repair of twocomponent parallel system with exponentially distributed lifetime by Boland and ElNeweihi [1].
In this paper our main aim is to compare two parallel systems in terms of reversed hazard rate order and likelihood ratio order with majorized scale and shape parameters separately, when the components are from two heterogeneous LL distributions as well as from the multiple outlier LL random variables. The rest of the paper is organized as follows. In Section 2, we have given the required notations, definitions and some useful lemmas which have been used throughout the paper. Results related to reversed hazard rate ordering and likelihood ratio ordering between two order statistics and are derived in Section 3.
Throughout the paper, the word increasing (resp. decreasing) and nondecreasing (resp. nonincreasing) are used interchangeably, and denotes the set of real numbers . We also write to mean that and have the same sign. For any differentiable function , we write to denote the first derivative of with respect to .
2 Notations, Definitions and Preliminaries
For an absolutely continuous random variable , we denote the probability density function, the distribution function and the reversed hazard rate function by and respectively. The survival or reliability
function of the random variable is written as .
In order to compare different order statistics, stochastic orders are used for fair and reasonable comparison.
In literature many different kinds of stochastic orders have been developed and studied.
The following well known definitions may be obtained in Shaked and Shanthikumar [19].
Definition 2.1
Let and be two absolutely continuous random variables with respective supports and , where and may be positive infinity, and and may be negative infinity. Then, is said to be smaller than in

likelihood ratio (lr) order, denoted as , if

hazard rate (hr) order, denoted as , if
which can equivalently be written as for all ;

reversed hazard rate (rhr) order, denoted as , if
which can equivalently be written as for all ;

usual stochastic (st) order, denoted as , if for all
In the following diagram we present a chain of implications of the stochastic orders, see, for instance, Shaked and Shanthikumar [19], where the definitions and usefulness of these orders can be found.
Definition 2.2
The vector is said to majorize the vector (written as ) if
Definition 2.3
A function is said to be Schurconvex (resp. Schurconcave) on if
Notation 2.1
Let us introduce the following notations.

.

.
Next, two lemmas are given which will be used to prove our main results. The first one can be obtained by combining Proposition H2 of Marshall et al. ([15], p. 132) and Lemma 3.2 of Kundu et al. ([9]) while the second one is due to Lemma 3.4 of Kundu et al. ([9]).
Lemma 2.1
Let with , where is differentiable, for all . Then is Schurconvex (Schurconcave) on if, and only if,
where .
Lemma 2.2
Let with , where is differentiable, for all . Then is Schurconvex (Schurconcave) on if, and only if,
where .
3 Main Results
For , let (resp. ) be independent nonnegative random variables following LL distribution as given in (1.1).
If and be the distribution functions of and respectively, where , , and , then
and
Again, if and are the reversed hazard rate functions of and respectively, then
(3.1) 
and
(3.2) 
The following two theorems show that under certain conditions on parameters, there exists reversed hazard rate ordering between and .
Theorem 3.1
For , let and be two sets of mutually independent random variables with and . Further, suppose that or . Then,
Proof: Let
giving
So, if , then Then, by Lemma 2.1 (Lemma 2.2), is Schur convex in , proving the result.
The counterexample given below shows that the ascending (descending) order of the components of the scale and shape parameters are necessary for the result of Theorem 3.1 to hold.
Counterexample 3.1
Let and Now, if , and are taken, then from Figure 1, it is clear that is not monotone, giving that , although .
Theorem 3.1 guarantees that for parallel systems of components having independent LL distributed lifetimes with common scale parameter vector, the majorized shape parameter vector leads to larger system’s life in the sense of the reversed hazard rate ordering. Now the question ariseswhat will happen if the scale parameter majorizes when the shape parameter vector remains constant? The theorem given below answers that if the order of the components of shape and scale parameter vectors are reversed, then will be smaller than in reversed hazard rate ordering.
Theorem 3.2
For , let and be two sets of mutually independent random variables with and . Further, suppose that , or , . Then,
Proof: For , let us consider Differentiating with respect to , we get
giving
So, if and , then So, by Lemma 2.1 (Lemma 2.2), is Schurconcave in , proving the result.
Next, one counterexample is provided to show that, nothing can be said about reversed hazard rate ordering between and if majorizes and all of , and are either in or in .
Counterexample 3.2
The following theorem shows that depending upon certain conditions, majorization order of the shape parameters implies likelihood ratio ordering between and .
Theorem 3.3
For , let and be two sets of mutually independent random variables with and . Further, suppose that or . Then, if ,
Proof:
In view of theorem 3.1 and using (3.1) and (3.2), here we have only to show that
is increasing in , where with respect to , . Now, differentiating
where
and
Thus, to show that is increasing in , we have only to show that
is Schurconcave in .
Now, as
and
then
So, if , i.e., for if and , then noticing the fact that is decreasing in as well as in it can be written that
Again, as implying for all , then
proving that is decreasing in . Again, it is also decreasing in . Thus, for all and ,
So, for all
Thus the result follows from Lemma 3.1 (Lemma 3.3) of Kundu et al. ([9]).
Although Theorem 3.3 holds under a sufficient condition for two component systems, the next theorem shows that no such condition is required for these systems having multipleoutlier LL model if the scale parameter vectors of these systems are common.
Theorem 3.4
For , let and be two sets of independent random variables each following the multipleoutlier EW model such that and for , and for If
and either or then .
Proof: Following Theorem 3.3 and in view of Theorem 3.1, we have only to show that
is Schurconcave in .
Now, three cases may arise:
If , , if and , then
If , , if and , then
If and , then , and , . It can be easily shown that
where and . Now, as and , implying that , and moreover, , then . Again, is increasing in
So, by Lemma 3.1 (Lemma 3.3) of Kundu et al. ([9]), the result is proved.
Theorem 3.3 guarantees that, for two component parallel systems (with a sufficient condition) having independent LL distributed lifetimes with a common scale parameter vector, the majorized shape parameter vector leads to greater system’s lifetime in the sense of likelihood ratio order. The next theorem states that the majorized scale parameter vector leads to smaller system’s lifetime in the sense of likelihood ratio order when the shape parameter vector of these two component parallel systems are common.
Theorem 3.5
For , let and be two sets of mutually independent random variables with and . Further, suppose that , or , . Then,
Proof: In view of Theorem 3.2 and using (3.1) and (3.2), we are to prove that
is decreasing in to prove that
is Schurconvex in . Now,
So, by noticing the fact that
giving that is increasing in , and , for all and gives
and
So,
Thus the result follows from Lemma 3.1 (Lemma 3.3) of Kundu et al. ([9]).
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