Many of the popular nonparametric test statistics for censored survival data used in two-sample, $k$-sample trend and continuous covariate situations are special cases of a general statistic, differing only in the choice of the covariate-based label and the weight function. A weight function determines the asymptotic efficiency of its corresponding statistic in this general class. Since the true alternatives are often unknown, we may not be able to foresee which weight function is the best for a particular data set. We show in this paper that certain large families of these statistics form stochastic processes, doubly indexed by both the weight function and the time scale, which converge weakly to Gaussian processes also indexed by both the weight function and the time scale. These asymptotic properties allow development of versatile test procedures which are simultaneously sensitive to a reasonably large collection of alternatives. Due to the complexity of the Gaussian processes, a Monte Carlo approach is proposed to obtain the distributional characteristics of these statistics under the null hypothesis.
"A general class of function-indexed nonparametric tests for survival analysis." Ann. Statist. 27 (5) 1722 - 1744, October 1999. https://doi.org/10.1214/aos/1017939149