# A Reproducing Kernel Hilbert Space log-rank test for the two-sample problem

## TL;DR

This paper introduces a novel two-sample test for right-censored data using a supremum of log-rank tests indexed by a reproducing kernel Hilbert space, offering robustness and simplicity.

## Contribution

It proposes a new RKHS-based log-rank test that is exact, easy to evaluate, and connects with existing methods, enhancing robustness against various alternatives.

## Key findings

- The test is omnibus for certain RKHS families.
- The method provides an exact evaluation of the test statistic.
- Empirical evaluation demonstrates its effectiveness.

## Abstract

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 weighted log-rank tests more robust against a broader family of alternatives, among them, considering linear combinations of weighted log-rank tests, and taking the maximum among a finite collection of them. In this paper, we propose as test statistic the supremum of a collection of (potentially infinite) weight-indexed log-rank tests where the index space is 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 finalise by performing an empirical evaluation of the proposed methodology and show an application to a real data scenario. Our theoretical results are proved using techniques for double integrals with respect to martingales that may be of independent interest.