On relative aging comparisons of coherent systems with identically distributed components
01 Jul 2021-Probability in the Engineering and Informational Sciences (Cambridge University Press)-Vol. 35, Iss: 3, pp 481-495
Abstract: The relative aging is an important notion which is useful to measure how a system ages relative to another one. Among the existing stochastic orders, there are two important orders describing the relative aging of two systems, namely, aging faster orders in the cumulative hazard and the cumulative reversed hazard rate functions. In this paper, we give some sufficient conditions under which one coherent system ages faster than another one with respect to the aforementioned stochastic orders. Further, we show that the proposed sufficient conditions are satisfied for k-out-of-n systems. Moreover, some numerical examples are given to illustrate the applications of proposed results.
...read more
Topics: Independent and identically distributed random variables (51%), Measure (mathematics) (51%)
Citations
More filters
TL;DR: This book aims to introduce simulation techniques for practitioners in the financial and risk management industry at an intermediate level by having extensive simulation examples using S–PLUS or Visual Basics.
...read more
Abstract: (2007). Stochastic Ageing and Dependence for Reliability. Technometrics: Vol. 49, No. 2, pp. 222-222.
...read more
294 citations
01 Jan 2021-
Abstract: In this paper, we investigate stochastic comparisons of the second largest order statistics of homogeneous samples coupled by Archimedean copula, and we establish the reversed hazard rate and likelihood ratio orders, and we further generalize the corresponding results to the case of random sample size. Also, we derive some results for relative ageing between parallel systems and $ 2 $-out-of-$ n $/$ 2 $-out-of-$ (n+1) $ systems. Finally, some examples are given to illustrate the obtained results.
...read more
16 Oct 2020-
Abstract: In this paper, the general proportional mean past lifetime frailty model is considered, from which the unconditional cumulative distribution and density functions of the lifetime variable are derived. Dependency between the two variables are studied. Stochastic comparisons are made through which it is shown that some well-known stochastic orderings between two frailty variables carry over to the corresponding lifetime variables. The effects of baseline variable and the frailty variable on the proposed frailty model are studied. The relative mean past lifetime ordering is introduced and some relative ordering between two lifetime random variables with different frailty variables are studied. Also a simulation study is given to illustrate some results.
...read more
TL;DR: This paper considers the systems that are formed by dependent and identically distributed components, where the dependency structures are described by Archimedean copulas, and investigates whether a system of used components performs better than a used system with respect to different stochastic orders.
...read more
Abstract:
Copula is one of the widely used techniques to describe the dependency structure between components of a system. Among all existing copulas, the family of Archimedean copulas is the popular one due to its wide range of capturing the dependency structures. In this paper, we consider the systems that are formed by dependent and identically distributed components, where the dependency structures are described by Archimedean copulas. We study some stochastic comparisons results for series, parallel, and general
$r$
-out-of-
$n$
systems. Furthermore, we investigate whether a system of used components performs better than a used system with respect to different stochastic orders. Furthermore, some aging properties of these systems have been studied. Finally, some numerical examples are given to illustrate the proposed results.
...read more
References
More filters
Book•
01 Jan 1999-
TL;DR: This book discusses the fundamental properties of copulas and some of their primary applications, which include the study of dependence and measures of association, and the construction of families of bivariate distributions.
...read more
Abstract: The study of copulas and their role in statistics is a new but vigorously growing field. In this book the student or practitioner of statistics and probability will find discussions of the fundamental properties of copulas and some of their primary applications. The applications include the study of dependence and measures of association, and the construction of families of bivariate distributions. This book is suitable as a text or for self-study.
...read more
8,616 citations
4,257 citations
Book•
01 Jun 1981-
TL;DR: A number of new classes of life distributions arising naturally in reliability models are treated systematically and each provides a realistic probabilistic description of a physical property occurring in the reliability context, thus permitting more realistic modeling of commonly occurring reliability situations.
...read more
Abstract: : This is the first of two books on the statistical theory of reliability and life testing. The present book concentrates on probabilistic aspects of reliability theory, while the forthcoming book will focus on inferential aspects of reliability and life testing, applying the probabilistic tools developed in this volume. This book emphasizes the newer, research aspects of reliability theory. The concept of a coherent system serves as a unifying theme for much of the book. A number of new classes of life distributions arising naturally in reliability models are treated systematically: the increasing failure rate average, new better than used, decreasing mean residual life, and other classes of distributions. As the names would seem to indicate, each such class of life distributions provides a realistic probabilistic description of a physical property occurring in the reliability context. Also various types of positive dependence among random variables are considered, thus permitting more realistic modeling of commonly occurring reliability situations.
...read more
3,824 citations
"On relative aging comparisons of co..." refers background in this paper
...A system is said to be coherent if each of its components is relevant, and its structure function is monotonically increasing in each argument (see [3])....
[...]
...For more discussion on this topic, the reader may go through Barlow and Proschan [3] and Lai and Xie [23]....
[...]
...F̄τ(X)(x) = P (τ (X) > x) = h ( F̄X(x) ) , x ∈ R, where the function h : [0, 1] → [0, 1] (called the domination (or dual distortion) function) depends on the structure function φ(·) (see [3] for definition) and on the survival copula K of X = (X1,X2, ....
[...]
...Further, the same problem, for general coherent systems, is considered in Esary and Proschan [12], Nanda et al. [29], Kochar et al. [21], Belzunce et al. [5], Navarro and Rubio [35], Navarro et al. [31–34], Navarro [30], Samaniego and Navarro [41], Amini-Seresht et al. [1], to name a few....
[...]
Book•
28 Mar 2006-
Abstract: This book provides a panoramic view of theory and applications of Ageing and Dependence in the use of mathematical methods in reliability and survival analysis. Ageing and dependence are important characteristics in reliability and survival analysis. They affect decisions with regard to maintenance, repair/replacement, price setting, warranties, medical studies, and other areas. Most of the works containing the topics covered here are theoretical in nature. However, this book offers applications, exercises, and examples. It serves as a reference for professors and researchers involved in reliability and survival analysis.
...read more
403 citations
TL;DR: This book aims to introduce simulation techniques for practitioners in the financial and risk management industry at an intermediate level by having extensive simulation examples using S–PLUS or Visual Basics.
...read more
Abstract: (2007). Stochastic Ageing and Dependence for Reliability. Technometrics: Vol. 49, No. 2, pp. 222-222.
...read more
294 citations
"On relative aging comparisons of co..." refers background in this paper
...For more discussion on this topic, the reader may go through Barlow and Proschan [3] and Lai and Xie [23]....
[...]