Hazard rate ordering of order statistics and systems (original) (raw)
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Tail hazard rate ordering properties of order statistics and coherent systems
Naval Research Logistics, 2007
We study tail hazard rate ordering properties of coherent systems using the representation of the distribution of a coherent system as a mixture of the distributions of the series systems obtained from its path sets. Also some ordering properties are obtained for order statistics which, in this context, represent the lifetimes of k-out-of-n systems. We pay special attention to systems with components satisfying the proportional hazard rate model or with exponential, Weibull and Pareto type II distributions.
Stochastic ordering properties for systems with dependent identically distributed components
Applied Stochastic Models in Business and Industry, 2013
In this paper, we obtain ordering properties for coherent systems with possibly dependent identically distributed components. These results are based on a representation of the system reliability function as a distorted function of the common component reliability function. So, the results included in this paper can also be applied to general distorted distributions. The main advantage of these results is that they are distribution-free with respect to the common component distribution. Moreover, they can be applied to systems with component lifetimes having a non-exchangeable joint distribution.
Characterization of some stochastic orders in reliability theory
2008
Various concepts of stochastic comparison between random variables have been de ned and studied in the literature, because of their usefulness in modeling for reliability, economics applications and as mathematical tools for proving important results in applied probability. Some well-known orders that have been introduced and studied in reliability theory are the hazard rate order, the total time on test transform order and the excess wealth order, whose de nitions are recalled in Chapters 1 and 2. As for stochastic orders, several non-parametric classes of life distributions have been proposed in the past decades in order to model different aspects of aging. The purpose of this thesis is provide some characterizations of the excess wealth order and the total time on test transform order. First, we consider a family of distribution functions and provide the relation between the hazard rate order and the excess wealth order. Second, we study the model of epoch times of non-homogeneous Poisson process via the excess wealth order. We also study further a parametrization of the excess wealth order and provide some characterizations for some classes of life distributions. On the other hand we give the relation between the dispersive order, the total time on test order and de ne the total time on test up to the kth order statistic. We show that the hazard rate order and the total time on test transform of residual lives order are actually equivalent dealing with continuous non-negative random variables and pointed out some consequences on the aging notions. We study further a parametrization of total time on test order and iii provide new characterization of the NBUT life distribution. Finally, we present the conclusion and future extensions.
University of the Free State Stochastic Ordering with Applications to Reliability Theory
2015
The rest of this note is organised into five chapters, which in turn divide into subsections. Chapter 1 discusses some of the most commonly used stochastic orders and their applications in Reliability. In Chapter 2 we discuss the ordering of the sub-family of Weibull distributions. Mixtures are presented in Chapter 3. This chapter also lays down the basics of what is presented in the penultimate chapter, which recalls some results on burn-in in heterogeneous populations. Finally we draw our conlusions in Chapter 5. v Chapter 1 An overview of common stochastic orders 1.1 Introduction We present herein, an overview of the most important univariate stochastic orders already common in the literature. By stochastic order we mean any rule that gives a probabilistic meaning to the inequality X Y where X and Y are (univariate) random variables. Stochastic orders have already been applied in Actuarial Science (e.g. in ordering risks) and Reliability (e.g. in ordering lifetimes). Usually in Reliability, one is concerned only with non-negative random variables. In the sequel, by lifetime we will mean a non-negative random variable and the terms will be used interchangeably. The majority of stochastic orders in the literature have the following desirable properties:
Some Results on Reversed Hazard Rate Ordering
Communications in Statistics: Theory and Methods, 2001
Recently, the reversed hazard rate (RHR) function, defined as the ratio of the density to the distribution function, has become a topic of interest having applications in actuarial sciences, forensic studies and similar other fields. Here we establish results with respect to RHR ordering between the exponentiated random variables. We also address the ordering results between component redundancy and system redundancy. Both the cases of matching spares and nonmatching spares are discussed. In case of matching spares, a sufficient condition has been given for component redundancy to be superior to the system redundancy with respect to the reversed hazard rate ordering for any coherent system.
Dependence Modeling, 2021
As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables T 1, ..., Tr. In any case, we assume that T 1, ..., Tr are identically distributed, with a common survival function ̄G and their survival copula is denoted by K. The diagonal sections of K, along with ̄G, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of T 1, ..., Tr also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects for the distrib...
Study of some measures of dependence between order statistics and systems
Journal of Multivariate Analysis, 2010
Let X = (X 1 , X 2 , . . . , X n ) be a random vector, and denote by X 1:n , X 2:n , . . . , X n:n the corresponding order statistics. When X 1 , X 2 , . . . , X n represent the lifetimes of n components in a system, the order statistic X n−k+1:n represents the lifetime of a k-out-of-n system (i.e., a system which works when at least k components work). In this paper, we obtain some expressions for the Pearson's correlation coefficient between X i:n and X j:n . We pay special attention to the case n = 2, that is, to measure the dependence between the first and second failure in a two-component parallel system. We also obtain the Spearman's rho and Kendall's tau coefficients when the variables X 1 , X 2 , . . . , X n are independent and identically distributed or when they jointly have an exchangeable distribution.
Mathematics, 2022
In this paper we focus on providing sufficient conditions for some well-known stochastic orders in reliability but dealing with the discrete versions of them, filling a gap in the literature. In particular, we find conditions based on the unimodality of the likelihood ratio for the comparison in some stochastic orders of two discrete random variables. These results have interest in comparing discrete random variables because the sufficient conditions are easy to check when there are no closed expressions for the survival functions, which occurs in many cases. In addition, the results are applied to compare several parametric families of discrete distributions.
On the reversed hazard rate of sequential order statistics
Statistics & Probability Letters, 2014
Sequential order statistics can be used to describe the lifetime of a system with n components which works as long as k components function assuming that failures possibly affect the lifetimes of remaining units. In this work, the reversed hazard rates of sequential order statistics are examined. Conditions for the reversed hazard rate ordering and the decreasing reversed hazard rate property of sequential order statistics are given.
Multivariate hazard rate orders
Journal of Multivariate Analysis, 2003
Two multivariate hazard rate stochastic orders are introduced and studied. Their meaning, properties, and relationship to other common stochastic orders are examined and investigated. Some examples that illustrate the theory are detailed. Finally, some applications of the new orders in reliability theory and in actuarial science are described. r