A reproducing kernel Hilbert space log-rank test for the two-sample problem

Weighted log-rank tests are arguably the most widely used tests by practitioners for the two-sample problem in the context of right-censored data. Many approaches have been considered to make them more robust against a broader family of alternatives, including taking linear combinations, or the maximum among a finite collection of them. In this article, we propose as test statistic the supremum of a collection of (potentially infinitely many) weighted log-rank tests where the weight functions belong to the unit ball in a reproducing kernel Hilbert space (RKHS). By using some desirable properties of RKHSs we provide an exact and simple evaluation of the test statistic and establish connections with previous tests in the literature. Additionally, we show that for a special family of RKHSs, the proposed test is omnibus. We finalize by performing an empirical evaluation of the proposed methodology and show an application to a real data scenario.