Bayesian Inference for P(X<Y) Using Asymmetric Dependent Distributions

Abstract

This paper studies Bayesian inference for $\theta =\mathbb{P}(X<Y)$ in the case where the marginal distributions of $X$ and $Y$ belong to classes of distributions obtained by skewing scale mixtures of normals. We separately address the cases where $X$ and $Y$ are independent or dependent random variables. Dependencies between $X$ and $Y$ are modelled using a Gaussian copula. Noninformative benchmark and vague priors are provided for these scenarios and conditions for the existence of the posterior distribution of $\theta$ are presented. We show that the use of the Bayesian models proposed here is also valid in the presence of set observations. Examples using simulated and real data sets are presented.