Summary: Many of the popular nonparametric test statistics for censored survival data used in two-sample, \(s\)-sample trend and single continuous covariate situations are special cases of a general statistic, differing only in the choice of covariate-based label and weight function. Formulated in terms of counting processes and martingales this general statistic, standardized by the square root of its consistent variance estimator, is shown to be asymptotically normal under the null hypothesis and under a sequence of contiguous hazard alternatives that includes both relative and excess risk models.
As an application to some specific cases of the general statistic, the asymptotic relative efficiencies of the B. W. Brown, M. Hollander and R. M. Korwar [in F. Proschan and R. J. Serfling (eds.), Reliability and biometry. Statistical analysis of lifelength. (1974; Zbl 0286.62071)] modification of the Kendall rank statistic, the Cox score statistic and the generalized logrank statistic of the authors [Biometrics 45, No. 1, 157-170 (1989; Zbl 0715.62086)] are investigated under relative and excess risk models. Finally, an example is given in which the Cox score test is not as efficient as the generalized logrank test in the presence of outliers in the covariate space.