Estimation of the Probability that Y < X

Abstract

The problem of estimating θ = Pr[Y < X] has been considered in the literature in both distribution-free and parametric frameworks. In this article, using a Bayesian approach, we consider the estimation of θ from two approaches. The first, analogous to the classical procedure, is concerned with the problem of parametric estimation. The second, peculiar to the Bayesian approach, is directed to the query, “For two future observations, × and Y, what is the probability (given only the available sample data) that Y is less than X” This probability, termed the predictive probability, is not an estimate but is, in fact, a probability. These two views are related in that this predictive probability is the mean of the posterior distribution of θ. In the following sections, these Bayesian procedures are applied to the case of independent exponentially distributed random variables and to various cases of the normal distribution. The Bayesian estimates thus obtained are compared, whenever possible, with their confidence counterparts, which are also derived here.