Ordering and aging properties of systems with dependent components governed by the Archimedean copula

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.

Ordering and ageing properties of developed sequential order statistics governed by the Archimedean copula

“…The corollary below follows from Theorem 3.2 and Remark 2.1. Note that the result given here is also mentioned in Sahoo and Hazra [37]. However, the set of sufficient conditions used in the latter paper is different from the one that is given here.…”

“…The corollary below follows immediately from Theorems 3.6 and Remark 2.1. Some special cases of this corollary are mentioned in Sahoo and Hazra [37].…”

“…(Sahoo and Hazra [37], Remark 3.1(a).) If is positive and increasing in , and is increasing in , then is positive and increasing in .…”

“…In the literature, numerous types of stochastic orders have been introduced, e.g., the usual stochastic order, the hazard rate order, etc. (see [36, 37]). Below we give the definitions of several stochastic orders that are used in subsequent sections.…”

“…One may note that all of the aforementioned studies were carried out for sequential k -out-of- n systems with independent components (or equivalently, sequential order statistics with independent random variables). The ordering and ageing properties of ordinary order statistics with dependent random variables, governed by the Archimedean copula, were considered in [28, 37] and the references therein. To the best of our knowledge, no work in this direction has been done for sequential order statistics with dependent random variables (i.e., for DSOS).…”