Asymptotic Behavior of Two-Sample Rank Tests in the Presence of Random Censoring

Abstract

Two samples, $\{X_{ji}, 1 \leq i \leq n(j)\} (j = 1, 2)$ are assumed to be composed of iid random variables with survival functions $(1 - F_j)(1 - H_j)$, where $H$ is the cdf of the "censoring times" and $F$ is the cdf of the "true lifetimes." A unified derivation of the Pitman efficiencies of a class of rank statistics for censored samples is presented. The conditions under which the result holds do not require contiguous alternatives, since convergence to normality is shown to hold uniformly in equicontinuous $(F_1, F_2, H_1, H_2)$ with bounded hazard rates. The uniformity is obtained by studying a convenient joint representation of several counting processes. The results are applied to the translated exponential distributions, a noncontiguous family of alternatives.