Preservation properties of stochastic orders by transformation to the transmuted-G model

Abstract

The transmuted-G model is a useful technique to construct some new distributions by adding a parameter. This paper considers stochastic comparisons in the transmuted-G family with different parameters and different baseline distributions in the sense of the usual stochastic, shifted stochastic, proportional stochastic and shifted proportional stochastic orders. Also, we present a necessary and sufficient condition for existence of the moments of the transmuted-G model and then we obtain some bounds for the survival and aging intensity functions of the transmuted-G model conditioned on its parameter and its baseline distribution.     

Necessary and sufficient conditions for stochastic orders between (n − r + 1)-out-of-n systems in proportional hazard (reversed hazard) rates model

Consider two (n ? r + 1)-out-of-n systems, one with independent and non-identically distributed components and another with independent and identically distributed components. When the lifetimes of components follow the proportional hazard rates model, we establish a necessary and sufficient condition for the usual stochastic order to hold between the lifetimes of these two systems. For the special case of r = 2, some generalized forms of this result to the hazard rate, dispersive and likelihood ratio orders are also obtained. Moreover, for the case when the lifetimes of components follow the proportional reversed hazard rates model, we derive some similar results for comparing the lifetimes of two systems . Applications of the established results to different situations are finally illustrated.     

Some ordering properties of series and parallel systems with dependent component lifetimes

Abstract

In this paper, we consider series systems and parallel systems with the dependence between the component lifetimes modelled by an Archimedean copulas. We obtain sufficient and necessary conditions of relative ageing orders between series (parallel) systems with different component numbers, which partially generalize some main results of Misra and Francis. When the component lifetimes follow the scale model, we also characterize the ordering properties between the series systems and (n–1)-out-of-n systems (parallel systems and 2-out-of-n systems) by mixture distribution.     

Ordering results for series and parallel systems comprising heterogeneous exponentiated Weibull components

In this paper, we discuss the usual stochastic and reversed hazard rate orders between the series and parallel systems from two sets of independent heterogeneous exponentiated Weibull components. We also obtain the results concerning the convex transform orders between parallel systems and obtain necessary and sufficient conditions under which the dispersive and usual stochastic orders, and the right spread and increasing convex orders between the lifetimes of the two systems are equivalent. Finally, in the multiple-outlier exponentiated Weibull models, based on weak majorization and p-larger orders between the vectors of scale and shape parameters, some characterization results for comparing the lifetimes of parallel and series systems are also established, respectively. The results of this paper can be used in practical situations to find various bounds for the important aging characteristics of these systems.     

New multivariate aging notions based on the corrected orthant and the standard construction

ABSTRACT

Recently, some well-known univariate aging classes of lifetime distributions have been characterized by means of properties of their quantile functions and excess-wealth functions. The generalization of the univariate aging notions to the multivariate case involve, among other factors, appropriate definitions of multivariate quantiles or regression representation and related notions, which are able to correctly describe the intrinsic characteristic of the concepts of aging that should be generalized. The multivariate versions of these notions, which are characterized by using the multivariate u-quantiles and the multivariate excess-wealth function, are considered in this paper. Relationships between such multivariate aging classes are studied, and examples are provided.     

Stochastic comparisons of two-unit repairable systems

Abstract

We consider two models of two-unit repairable systems: cold standby system and warm standby system. We suppose that the lifetimes and repair times of the units are all independent exponentially distributed random variables. Using stochastic orders we compare the lifetimes of systems under different assumptions on the parameters of exponential distributions. We also consider a cold standby system where the lifetimes and repair times of its units are not necessarily exponentially distributed.     

Stochastic orders and aging notions based upon the moment generating function order: Theory

The purpose of this paper is to study new notions of stochastic comparisons and aging notions based on the moment generating function order. Two characteristics of the moment generating function order of residual lives are obtained, and some closure properties of the aging classes DRL_mg (IRL_mg), NBU_mg (NWU_mg) and EBU_mg (EWU_mg) under some transformations are developed. As application, some results for random sums with applications to shock models are obtained.     

Exponentiated models preserve stochastic orderings of parallel and series systems

Abstract

In this paper, we establish that the usual stochastic, hazard rate, reversed hazard rate, likelihood ratio, dispersive and star orders are all preserved for parallel systems under exponentiated models for lifetimes of components. We then use the multiple-outlier exponentiated gamma models to illustrate this result. Finally, we consider the dual family with exponentiated survival function and establish similar results for series systems. The results established here extend some well-known results for series and parallel systems arising from different exponentiated distributions such as generalized exponential and exponentiated Weibull, established previously in the literature.     

The percentile residual life up to time t0: Ordering and aging properties

Motivated by practical issues, a new stochastic order for random variables is introduced by comparing all their percentile residual life functions until a certain instant. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are studied and also an application in reliability theory is described. Finally, we present some characterization results of the decreasing percentile residual life up to time t₀ aging notion.     

Dynamic proportional hazard rate and reversed hazard rate models

Cox (1972) proportional hazard (PH) model has been used to model failure time data in Reliability and Survival Analysis. Recently, proportional reversed hazard model has been analyzed in the literature. Sometimes, the hazard rate (or the reversed hazard rate) may not be proportional over the whole time interval, but may be proportional differently in different intervals. In order to take care of this kind of problems, in this paper, we introduce the dynamic proportional hazard rate model, and the dynamic proportional reversed hazard rate model, and study their properties for different aging classes. The closure of the models under different stochastic orders has also been studied. Examples are presented to illustrate different properties of the models.     

Further results on discrete mean past lifetime

Abstract

The present paper aims at studying the mean past lifetime of a discrete random variable. The notion of discrete mean past lifetime is studied in relation to the concepts of reversed hazard rate, reversed lack of memory property, and cumulative past entropy. New classes of distributions characterized by particular forms of discrete mean past life are also investigated. Implications of an increasing mean past lifetime on other reliability notions are studied and finally some bivariate generalizations are discussed.     

Some characterization results for parallel systems with two heterogeneous exponential components

In this paper, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components in terms of the likelihood ratio order (reversed hazard rate order) and the hazard rate order (stochastic order). We establish, among others, that the weakly majorization order between two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between lifetimes of two parallel systems, and that the p-larger order between two hazard rate vectors is equivalent to the hazard rate order (stochastic order) between lifetimes of two parallel systems. Moreover, we extend the results to the proportional hazard rate models. The results derived here strengthen and generalize some of the results known in the literature.     

A Mixture Model of Proportional Reversed Hazard Rate

This article investigates properties of mixture model of proportional reversed hazard rate. Firstly, the mixing random variable and the overall population variable are proved to be positively likelihood dependent. Secondly, lower bounds for the distribution function as well as the conditional distribution are established in the case that the mixing variable belongs to certain nonparametric classes. Finally, some stochastic orders on the mixing (baseline) variables are proved to be translated to the corresponding overall population variables.     

Further results for parallel systems with two heterogeneous exponential components

In this paper, we investigate ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components with respect to likelihood ratio and hazard rate orders. Two sufficient conditions are provided for likelihood ratio and hazard rate orders to hold between the lifetimes of two parallel systems, respectively. Moreover, we extend the results from exponential case to the proportional hazard rate models. The results established here strength some of the results known in the literature. Finally, some numerical examples are given to illustrate the theoretical results derived here as well.     

On the effect of dependency in information properties of series and parallel systems

Information measures of reliability systems has been widely studied in the statistical and reliability literatures. These findings were obtained when lifetimes of components are independent and identically distributed. But, there is no context about the information properties of such systems when lifetimes of components are dependent. In this paper, we explore properties of the entropy and Kullback–Leibler discrimination information for series and parallel system’s lifetimes when lifetimes of components are dependent and identically distributed. Specifically, we give some results on the entropy of series systems when lifetimes of components are positive or negative quadrant dependence. Moreover, several results are obtained about the entropy ordering properties related to other well known stochastic orders. To illustrate the quality of the given results, some examples are also given.     

Bivariate Aging Properties under Archimedean Dependence Structures

Let X  = (X, Y) be a pair of lifetimes whose dependence structure is described by an Archimedean survival copula, and let X _t  = [(X ? t, Y ? t) | X > t, Y > t] denotes the corresponding pair of residual lifetimes after time t ≥ 0. Multivariate aging notions, defined by means of stochastic comparisons between X and X _t , with t ≥ 0, were studied in Pellerey (2008 Pellerey , F. ( 2008 ). On univariate and bivariate aging for dependent lifetimes with Archimedean survival copulas . Kybernetika 44 : 795 – 806 .[Web of Science ®] , [Google Scholar]), who considered pairs of lifetimes having the same marginal distribution. Here, we present the generalizations of his results, considering both stochastic comparisons between X _t and X _t+s for all t, s ≥ 0 and the case of dependent lifetimes having different distributions. Comparisons between two different pairs of residual lifetimes, at any time t ≥ 0, are discussed as well.     

On a system of nonhomogeneous components sharing a common frailty

The components of a reliability system subjected to a common random environment usually have dependent lifetimes. This paper studies the stochastic properties of such a system with lifetimes of the components following multivariate frailty models and multivariate mixed proportional reversed hazard rate (PRHR) models, respectively. Through doing stochastic comparison, we devote to throwing a new light on how the random environment affects the number of working components of a reliability system and on assessing the performance of a k-out-of-n system.     

On the Mixture Proportional Mean Residual Life Model

In this article, we propose a new mixture model induced by the model of proportional mean residual life. Under some appropriate assumptions, it is shown that the mixing and overall variables in the model admit the positive likelihood ratio dependence structure. To see how the overall variable is affected by the stochastic variation of the mixing variable, we study some stochastic comparisons using these variables. Finally, some useful bounds for tail probability of the overall variable for large values of the mixing variable are derived.     

Weighted extensions of generalized cumulative residual entropy and their applications

Abstract

In this paper, we consider weighted extensions of generalized cumulative residual entropy and its dynamic(residual) version. Our results include linear transformations, stochastic ordering, bounds, aging class properties and some relationships with other reliability concepts. We also define the conditional weighted generalized cumulative residual entropy and discuss some properties of its. For these concepts, we obtain some characterization results under some assumptions. Finally, we provide an estimator of the new information measure using empirical approach. In addition, we study large sample properties of this estimator.     

Sequential order statistics from dependent random variables

Abstract

This paper provides an extension for “sequential order statistics” (SOS) introduced by Kamps. It is called “developed sequential order statistics” (DSOS) and is useful for describing lifetimes of engineering systems when component lifetimes are dependent. Explicit expressions for the joint density function, the marginal distributions and the means of DSOS are derived. Under the well known “conditional proportional hazard rate” (CPHR) model and the Gumbel families of copulas for dependency among component lifetimes, some findings are reported. For example, it is proved that the joint density functions of DSOS and SOS have the same structure. Various illustrative examples are also given.