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

We address the statistical problem of evaluating R = P(X < 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.