Estimation of the reliability parameter for a Poisson-exponential stress-strength model

In this paper, we consider the problem of estimating the reliability parameter of a mixed-type stress-strength model, i.e., the probability R=P(X<Y) where X and Y are a discrete and a continuous random variable, respectively. We focus on the specific case of Poisson stress and exponential strength, deriving the expression of R and its maximum likelihood estimator (MLE) and its uniformly minimum-variance unbiased estimator (UMVUE), based on simple random samples independently drawn from X and Y. For the MLE, we are able to provide an expression for the cumulative distribution function, which allows us to compute its expected value, bias, and variance. We derive asymptotic properties of the MLE, which we exploit for constructing approximate confidence intervals based on different approaches. A simulation study empirically compares such estimators and provides advice for their correct use, which is also illustrated through an application to real data.