Connecting copula properties with reliability properties of coherent systems

Navarro, Jorge; Durante, Fabrizio; Sánchez, Juan Fernández

doi:10.1002/asmb.2579

Cited by 6 publications

(6 citation statements)

References 27 publications

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“…Thus, E(X) − E(X|Y > y) and E(X|Y ≤ y) − E(X) must have the same sign, and the equivalence follows. In a similar manner, again by using Inequality (16), one can prove the equivalence between (i) and (iii) (see [20], Theorem 3.1, for details). For the relationship between (i) and (iv), observe that:…”

Section: Corollarymentioning

confidence: 87%

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Weak Dependence Notions and Their Mutual Relationships

2020

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New weak notions of positive dependence between the components X and Y of a random pair (X,Y) have been considered in recent papers that deal with the effects of dependence on conditional residual lifetimes and conditional inactivity times. The purpose of this paper is to provide a structured framework for the definition and description of these notions, and other new ones, and to describe their mutual relationships. An exhaustive review of some well-know notions of dependence, with a complete description of the equivalent definitions and reciprocal relationships, some of them expressed in terms of the properties of the copula or survival copula of (X,Y), is also provided.

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Section: Corollarymentioning

confidence: 87%

“…Some of these relationships were given in [13], p. 166, and in [14]. Connections with the ordering properties of coherent systems and extremes values were given in [16,17]. Table 2.…”

Section: Definitionmentioning

confidence: 99%

Weak Dependence Notions and Their Mutual Relationships

2020

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“…Here µ can be estimated from the training sample. Another option to estimate U from L is to use the conditional median curve m2|1 (x) := G −1 2|1 (0.5|x) that can be obtained from (17) and the inverse function of D 2|1 (u|F (x)). Even more, we can obtain quantile-confidence bands (see Section 2.4).…”

Section: A Uniform Distribution Over This Triangle)mentioning

confidence: 99%

“…The joint distribution of two semi-coherent systems with some shared components was studied in [12,19]. Connections between dependence (copula) properties and ordering properties of systems were studied in [17].…”

Section: Coherent Systemsmentioning

confidence: 99%

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Distortion Representations of Multivariate Distributions

Navarro¹,

Calì²,

Longobardi³

et al. 2020

Preprint

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The univariate distorted distribution were introduced in risk theory to represent changes (distortions) in the expected distributions of some risks. Later they were also applied to represent distributions of order statistics, coherent systems, proportional hazard rate (PHR) and proportional reversed hazard rate (PRHR) models, etc. In this paper we extend this concept to the multivariate setup. We show that, in some cases, they are a valid alternative to the copula representations especially when the marginal distributions may not be easily handled. Several relevant examples illustrate the applications of such representations in statistical modeling. They include the study of paired (dependent) ordered data, joint residual lifetimes, order statistics and coherent systems.

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Comparisons of coherent systems with two types of heterogeneous components having proportional reversed hazard rates

Rao,

Naqvi

2023

Appl Stoch Models Bus & Ind

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The comparison of coherent systems in terms of stochastic orders is vital in reliability theory. While there is a considerable amount of literature devoted to comparing systems with homogeneous and independent components, real‐world systems often consist of heterogeneous components. Hence, this article aims to investigate systems with heterogeneous and independent components, as well as, those with heterogeneous and dependent components. For this purpose, we consider systems comprise of three components, which are of two different types of components, namely two components of type A and one component of type B. The system's lifetime distribution is represented using the failure signature when the components are independent, which is a function of the component's life distribution. However, when the components are dependent, the system's lifetime distribution is represented using copula and diagonal sections. Additionally, distorted distributions are utilized to enable distribution‐free stochastic comparisons. Using these representations, we compare systems with components having proportional reversed hazard rates, in three scenarios: (i) when components are heterogeneous and independent; (ii) when components are heterogeneous and dependent; and finally, (iii) comparing systems with homogeneous and independent components with those that have heterogeneous components. To illustrate the applicability of these results, we provide some examples and applications.

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Connecting copula properties with reliability properties of coherent systems

Cited by 6 publications

References 27 publications

Weak Dependence Notions and Their Mutual Relationships

Weak Dependence Notions and Their Mutual Relationships

Distortion Representations of Multivariate Distributions

Comparisons of coherent systems with two types of heterogeneous components having proportional reversed hazard rates

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