Conditional Rank Tests for the Two-Sample Problem Under Random Censorship

Abstract

For the two-sample problem with randomly censored data, there exists a general asymptotic theory of rank statistics which are functionals of stochastic integrals with respect to certain empirical martingales. In the present paper a conditional counterpart of this theory is developed. The conditional martingales are versions of the original ones reduced to the unit interval having their jumps at fixed lattice points. The resulting conditional tests are strictly distribution free under the null hypothesis of randomness if the censoring distributions in both samples are equal and are asymptotically equivalent to their unconditional counterparts even if the censoring distributions are different. Simulations for linear rank statistics and Kolmogorov-Smirnov-type statistics show superiority of the conditional versions over their unconditional counterparts with respect to size and robustness under unequal censoring in both samples. At the same time the power of the conditional and unconditional tests is very similar in most cases.