On an inverse problem in mixture failure rates modelling
type: Item
instance of: scholarly article
title: On an inverse problem in mixture failure rates modelling
zbMATH Open document ID: 0973.62090
author: M. S. Finkelstein
author: Veronica Esaulova
published in: Applied Stochastic Models in Business and Industry
publication date: 2001-09-16T00:00:00Z
Mathematics Subject Classification ID: 62N05
Mathematics Subject Classification ID: 60K10
zbMATH DE Number: 1646426
zbMATH Keywords: mixtures
zbMATH Keywords: failure rates
zbMATH Keywords: additive model
zbMATH Keywords: multiplicative model
MaRDI profile type: Publication
full work available at URL: asmb.435
OpenAlex ID: W2166809310
Wikidata QID: Q58293389
cites work: Tail behaviour of the failure rate functions of mixtures
cites work: A Bayes Explanation of an Apparent Failure Rate Paradox
DOI: 10.1002/ASMB.435
review text: This article deals with the following inverse problem in mixture failure rates modelling: Given the mixture failure rate and the mixing distribution, obtain the failure rate of the governing distribution. Let \(F(t,\theta)\) be a governing distribution for a fixed \(\theta\) of the continuous lifetime random variable \(T\geq 0\), and let \(\theta\) be a non-negative continuous random variable with probability density function \(\pi(\theta)\). Let \(\pi(\theta |t)\) denote the conditional probability density function of \(\theta\) given \(T\geq t\), then the mixture failure rate \(\lambda_{m}(t)\) can be defined as NEWLINE\[NEWLINE\lambda_{m}(t)=\int_{0}^{\infty}\lambda(t,\theta) \pi(\theta |t)d\theta,NEWLINE\]NEWLINE where \(\lambda(t,\theta)\) is the failure rate of the governing distribution. The authors consider additive \(\lambda(t,\theta)= \alpha(t)+\theta\) and multiplicative \(\lambda(t,\theta)=\theta\alpha(t)\) models of mixing, where \(\alpha(t)\) is some baseline failure rate function. It is shown that for these models the inverse problem can be solved, which means that given an arbitrary shape of the mixture failure rate and mixing distribution the failure rate for the governing distribution can be uniquely obtained.
reviewed by: Aleksandr D. Borisenko
description: scientific article; zbMATH DE number 1646426