On an inverse problem in mixture failure rates modelling (Q2739986)

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.