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.

中文翻译：

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两样本问题的再现核希尔伯特空间对数秩检验

在右删失数据的背景下，加权对数秩检验可以说是实践者针对双样本问题使用最广泛的检验。许多方法被认为可以使它们对更广泛的替代方案更加稳健，包括采用线性组合，或它们的有限集合中的最大值。在本文中，我们建议将一组（可能无限多）加权对数秩检验的上确界作为检验统计量，其中权重函数属于再现核希尔伯特空间 (RKHS) 中的单位球。通过使用 RKHS 的一些理想特性，我们提供了对测试统计量的准确和简单的评估，并与文献中的先前测试建立了联系。此外，我们表明，对于一个特殊的 RKHS 家族，建议的测试是综合性的。