Hazard rate ordering of order statistics and systems

Rubio

2010

TEST

Self Cite

The comparisons of coherent systems is a relevant topic in reliability and survival studies. To this purpose, several techniques have been proposed including comparisons of expected lifetimes or comparisons using stochastic orderings. Recently, Samaniego (System Signatures and Their Applications in Engineering Reliability. ) proposed using stochastic precedence to compare the lifetimes of two independent coherent systems. He proved that if the components in both systems are independent and identically distributed (IID), then these comparisons are distribution free, while comparisons based on expected values are not. In the present paper, we obtain new expressions to compare systems using stochastic precedence without the assumption of IID components. In particular, we show that if the components in both systems are independent and satisfy the Cox proportional hazard rate (PHR) model with the same baseline hazard rate, then these comparisons do not depend on the baseline hazard rate function.

Section: Applicationssupporting

confidence: 92%

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Comparisons of coherent systems using stochastic precedence

Rubio

2010

TEST

Self Cite

“…In the next proposition, we show that if the system T is the tail best system in the HR order (in the sense of (3.4) below), then its reliability function is tail equivalent to that of the average system. Moreover, from Lemma 3.1 in Navarro and Shaked [18] (see also [19]), we have that, if (3.4) holds for π , then lim t→∞ F T π (t)…”

Section: Definition 31mentioning

confidence: 92%

“…Then, by applying Lemma 3.3 in Navarro and Shaked [18] (see also [19]) to the mixture form in (3.9), we have that, if (3.4) and (3.7) hold for all π ∈ C 2 , then…”

Section: Proposition 33 If T Is the Lifetime Of A Coherent System Wmentioning

confidence: 99%

Applications of average and projected systems to the study of coherent systems

Journal of Multivariate Analysis

Spizzichino

Balakrishnan

2010

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In this paper, we introduce the concepts of average and projected systems associated to a coherent (parent) system. We analyze several aspects of these notions and show that they can be useful tools in studying the performance of coherent systems with non-exchangeable components. We show that the average and projected systems are especially useful in studying the tail behavior of reliability, hazard rate and mean residual life functions of the parent system and also in obtaining the tail best systems (under different criteria) by permuting the components at the system structure. Moreover, they can be useful in assessing how the asymmetry of the joint distribution of the component lifetimes (with respect to permutations of the components in the system structure) affects the system performance. (C) 2010 Elsevier Inc. All rights reserved

Stochastic ordering properties for systems with dependent identically distributed components

Appl Stoch Models Bus & Ind

Aguila

Sordo

et al. 2012

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115

106

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. Appl. Stochastic ModelsBus. Ind. 2013, 29 264-278 J. NAVARRO ET AL.The rest of the paper is organized as follows. In Section 2, we give the main results of the paper with the general ordering results for distorted distributions and coherent systems with ID components. Specifically, we study properties of the usual stochastic order, hazard rate order, the reversed hazard rate order, the likelihood ratio order, the increasing convex order, the total time on test transform order, and the excess wealth order. In Section 3, we give some examples to show how to apply our general results to specific systems. These examples include systems with IID components and systems with dependent ID components. In Section 4, we give some conclusions and open questions for future research.Throughout the paper, we use the terms increasing and decreasing in a wide sense, that is, a function g is increasing (decreasing) if g.x/ 6 g.y/.>/ for all x 6 y. Whenever we consider an expectation (or a conditional random variable), we assume that it exists. The main resultsLet us consider a coherent system with lifetime T D .X 1 ; : : : ; X n / based on possibly dependent components with lifetimes X 1 ; : : : ; X n , where is the structure function (see [29], Chapter 1). Let us assume that X 1 ; : : : ; X n are identically distributed (ID) with a common reliability function F .t/ D Pr.X i > t/ for i D 1; : : : ; n. The component lifetimes X 1 ; : : : ; X n can be dependent, and this dependence will be represented by the joint reliability (or survival) function of .X