Approximations and two-sample tests based on P− P and Q− Q plots of the Kaplan-Meier estimators of lifetime distributions

Abstract

Let F n and G n denote the Kaplan-Meier product-limit estimators of lifetime distributions based on two independent samples, and let F n inv and G n inv denote their quantile functions. We consider the corresponding P− P plot F n ( G n inv) and Q− Q plot F n inv( G n ), and establish strong approximations of empirical processes based on these P− P and Q− Q plots by appropriate sequences of Gaussian processes. It is shown that the rates of approximation we obtain are the best which can be achieved by this method. We apply these results to obtain the limiting distributions of test statistics which are functionals of F n ( G n inv( s)) − s, G n ( F n inv( s)) − s, and F n ( G n inv( s)) + G n ( F n inv( s)) − 2 s, and propose solutions to the problem of testing the assumption that the underlying lifetime distributions F and G are equal, in the case where the censoring distributions are arbitrary and unknown.