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Journal ArticleDOI

On relative aging comparisons of coherent systems with identically distributed components

TL;DR: In this article, sufficient conditions under which one coherent system ages faster than another one with respect to the aforementioned stochastic orders are given. And the proposed sufficient conditions are satisfied for k-out-of-n systems.
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.
Citations
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Journal ArticleDOI
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.
Abstract: (2007). Stochastic Ageing and Dependence for Reliability. Technometrics: Vol. 49, No. 2, pp. 222-222.

314 citations

Journal ArticleDOI
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.
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.

2 citations

Journal ArticleDOI
TL;DR: In this article , a general characterization property considering two new dynamic relative reliability measures is obtained, which are expressed as the ratio of hazard rates and as a ratio of reversed hazard rates, and evaluated partially at some sequential random times following a specific distribution.
Abstract: In this paper, a general characterization property considering two new dynamic relative reliability measures is obtained. The new dynamic relative reliability measures are expressed as the ratio of hazard rates and as the ratio of reversed hazard rates. The measures are evaluated partially at some sequential random times following a specific distribution. We show that several particular statistics, as random times, fulfill that specific distribution, and thus, the result is applicable in the context of the specified random times. The results are applied to some examples to characterize the Weibull distribution and the inverse Weibull distribution.

1 citations

Journal ArticleDOI
16 Oct 2020
TL;DR: In this article, the unconditional cumulative distribution and density functions of the lifetime variable are derived from a general proportional mean past lifetime frailty model, and the effects of the baseline variable and the frailty variable on the proposed model are studied.
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.

1 citations

Journal ArticleDOI
01 Jan 2021
TL;DR: In this article, the second largest order statistics of homogeneous samples coupled by Archimedean copula were investigated, and the reversed hazard rate and likelihood ratio orders were established.
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.
References
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Book
06 Oct 2015
TL;DR: An Introduction to Stochastic orders discusses the powerful tool that can be used in comparing probabilistic models in different areas such as reliability, survival analysis, risks, finance, and economics as discussed by the authors.
Abstract: An Introduction to Stochastic Orders discusses this powerful tool that can be used in comparing probabilistic models in different areas such as reliability, survival analysis, risks, finance, and economics. The book provides a general background on this topic for students and researchers who want to use it as a tool for their research. In addition, users will find detailed proofs of the main results and applications to several probabilistic models of interest in several fields, and discussions of fundamental properties of several stochastic orders, in the univariate and multivariate cases, along with applications to probabilistic models. * Introduces stochastic orders and its notation* Discusses different orders of univariate stochastic orders* Explains multivariate stochastic orders and their convex, likelihood ratio, and dispersive orders

118 citations

Journal ArticleDOI
TL;DR: Methods of survival analysis for long-term follow-up studies are illustrated by a study of mortality in 3878 breast cancer patients in Edinburgh followed for up to 20 years, and an additive hazard model is developed to incorporate covariates in the analysis of relative survival and curability.
Abstract: Methods of survival analysis for long-term follow-up studies are illustrated by a study of mortality in 3878 breast cancer patients in Edinburgh followed for up to 20 years. The problems of life tables, advantages of hazard plots and difficulties in statistical modelling are demonstrated by studying the relationship between survival and both clinical stage and initial menopausal status at diagnosis. To assess the ‘curability’ of breast cancer, mortality by year of follow-up is compared with expected mortality using Scottish age-specific death rates. Techniques for analysing such relative survival data include age-corrected life tables, ratio of observed to expected deaths and excess death rates. Finally, an additive hazard model is developed to incorporate covariates in the analysis of relative survival and curability.

110 citations


"On relative aging comparisons of co..." refers background in this paper

  • ...[38] noticed the crossing hazards phenomenon when they were dealing with the survival data on the effects of two different treatments to breast cancer patients....

    [...]

Journal ArticleDOI
TL;DR: In this article, the authors obtained ordering properties for coherent systems with possibly dependent identically distributed components based on a representation of the system reliability function as a distorted function of the common component reliability function.
Abstract: 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. Copyright © 2012 John Wiley & Sons, Ltd.

108 citations


"On relative aging comparisons of co..." refers background in this paper

  • ...[31]): Let τ(X) be the lifetime of a coherent system formed by n d....

    [...]

Journal ArticleDOI
TL;DR: The proportional reversed hazards model as mentioned in this paper describes random failure times by a family of distribution functions, where F(x) is a baseline distribution function and X is a random distribution function.

102 citations


"On relative aging comparisons of co..." refers background in this paper

  • ...Later, Di Crescenzo [9] developed two relative aging notions, one based on monotonicity of the ratio of two reversed hazard functions and another one based on monotonicity of the ratio of two cumulative reversed hazard functions....

    [...]

Journal ArticleDOI
TL;DR: In this article, the concepts of decreasing mean residual life, new better than used in expectation, harmonic new better, and new better in failure rate average are generalized, so as to compare the aging properties of two arbitrary life distributions.
Abstract: New partial orderings of life distributions are given. The concepts of decreasing mean residual life, new better than used in expectation, harmonic new better than used in expectation, new better than used in failure rate, and new better than used in failure rate average are generalized, so as to compare the aging properties of two arbitrary life distributions.

89 citations