Recent advances on Bayesian inference for $P(X < Y)$

Abstract

We address the statistical problem of evaluating $R = P(X \lt Y)$, where $X$ and $Y$ are two independent random variables. Bayesian parametric inference is based on the marginal posterior density of $R$ and has been widely discussed under various distributional assumptions on $X$ and $Y$. This classical approach requires both elicitation of a prior on the complete parameter and numerical integration in order to derive the marginal distribution of $R$. In this paper, we discuss and apply recent advances in Bayesian inference based on higher-order asymptotics and on pseudo-likelihoods, and related matching priors, which allow one to perform accurate inference on the parameter of interest $R$ only, even for small sample sizes. The proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals. From a theoretical point of view, we show that the used prior is a strong matching prior. From an applied point of view, the accuracy of the proposed methodology is illustrated both by numerical studies and by real-life data concerning clinical studies.