Alternative Estimation Procedures for Pr(X < Y) in Categorized Data

Abstract

Consider two independent random variables X and Y. The functional R = Pr(X < Y) [or .lambda. = Pr(X < Y)-Pr(Y < X)] is of practical importance in many situations, including clinical trials, genetics, and reliability. In this paper several approaches to estimation of .lambda. when X and Y are presented in discretized (categorical) form are analyzed and compared. Asymptotic formulas for the variances of the estimators are derived; use of the bootstrap to estimate variances is also discussed. Computer simulations indicate that the choice of the best estimator depends on the value of .lambda., the underlying distribution, and the sparseness of the data. It is shown that the bootstrap provides a robust estimate of variance. Several examples are treated.