Combined rank tests for the two-sample problem with randomly censored data

In the two-sample problem under random censorship we consider the uncensored observations and the censored ones as two separate groups. For each group a suitable rank statistic is obtained, and these two are then combined to a final one. This idea, which was first employed in Akritas (1983), is shown here to produce tests that (a) closely resemble the ordinary rank tests for the uncensored case and do not require the calculation of the Kaplan-Meier estimator, (b) are comparatively easy to apply and to understand, and (c) allow results on asymptotic normality to follow simply from standard results for the uncensored case. It is shown that the loss due to using two separate rankings rather than one complete ranking is asymptotically negligible. The optimal score function for each of the two separate rank statistics is seen to depend on the censoring distribution. Whereas in Akritas (1983) no assumption on the form of the censoring distribution was made (unrestricted adaptation), here we pursue restricted adaptation that results in simple score functions. In particular, motivated by the model of Koziol and Green (1976), we assume that the censoring distribution is a suitable function of the survival distribution. An example related to the Wilcoxon test is used for illustration. The effect of restricted adaptation on the efficiency of the test is examined by both exact efficiency calculations and simulation results.