Multivariate Reversed Hazard Rates and Inactivity Times of Systems

Francesco Buono (), Emilio Santis (), Maria Longobardi () and Fabio Spizzichino ()
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Francesco Buono: Università degli Studi di Napoli Federico II
Emilio Santis: Università La Sapienza
Maria Longobardi: Università degli Studi di Napoli Federico II
Fabio Spizzichino: Università La Sapienza

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 3, 1987-2008

Abstract: Abstract The family of the multivariate conditional hazard rate functions often reveals to be a convenient tool to describe the joint probability distribution of a vector of non-negative random variables (lifetimes) in the absolutely continuous case. Such a tool can have in particular an important role in the study of the behavior of the minima among inter-dependent lifetimes. In this paper we introduce the concept of reversed multivariate conditional hazard rate functions, which extends the one-dimensional notion of reversed hazard rate of a single non-negative random variable. Several basic properties of this concept are proven. In particular, we point out a related role in the study of the behavior of the maximum value among inter-dependent lifetimes. In different applied fields, and in particular in the reliability literature, a remarkable class of dependence models for vectors of lifetimes is related with the load-sharing condition, which can be defined in terms of the multivariate conditional hazard rate functions. In the paper we define the class of reversed load-sharing models, which can be seen as natural extensions to the multivariate case of the univariate inverse exponential distributions. We analyze basic properties of such a class of dependence models. In particular we show a result related to the study of the inactivity time of a coherent system when the joint distribution of the components’ lifetimes is a reversed load-sharing model.

Keywords: Distributions of maximum among dependent variables; Inverse exponential distributions; Reverse Load-Sharing models; Inactivity times of systems; 62H05; 62N05; 60E99 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11009-021-09905-2

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