Log-Rank-Type Tests for Equality of Distributions in High-Dimensional Spaces

Motivated by applications in high-dimensional settings, we propose a novel approach for testing the equality of two or more populations by constructing a class of intensity centered score processes. The resulting tests are analogous in spirit to the well-known class of weighted log-rank statistics that are widely used in survival analysis. The test statistics are nonparametric, computationally simple, and applicable to high-dimensional data. We establish the usual large sample properties by showing that the underlying log-rank score process converges weakly to a Gaussian random field with zero mean under the null hypothesis and with a drift under the contiguous alternatives. For the Kolmogorov-Smirnov-type and the Cramér-von Mises-type statistics, we also establish the consistency result for any fixed alternative. Cutoff points for the test statistics are obtained by permutations or a simulation-based resampling method. The new approach is applied to a study of brain activation measured by functional magnetic resonance imaging when performing two linguistic tasks and also to a prostate cancer DNA microarray data set.