Random inequalities of the form Prob (x > y + δ) Prob often appear as part of Bayesian clinical trial methods. Simulating trial designs could require calculating millions of random inequalities. When these inequalities require numerical integration, or worse random sampling, the inequality calculations account for the large majority of the simulation time. In this paper we show how to approximate random inequalities using Edgeworth expansions. The calculations required to use these expansions can be done in closed form, as we will see below. Although the calculations are elementary, they are also somewhat tedious, and so we include Python code to illustrate how to use the approximations in practice. We make no distributional assumptions on the random variables X and Y other than requiring that the necessary moments exist. The accuracy of the approximation will depend on how well the densities of these random variables are approximated by the Edgeworth expansions.
Cook, John D., "Approximating random inequalities with Edgeworth expansions" (November 2012). UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series. Working Paper 78.