Bernstein–based Nonparametric Estimation of the Cross Ratio Function under Univariate Right Censoring

Abstract

Bivariate time-to-event data often arise in various fields, including medicine, engineering, and economics, where understanding the association between two survival times is crucial. Traditional global association measures like Spearman's rho and Kendall's tau provide an average assessment, but fail to capture how association evolves over time. Local association measures, on the other hand, including the so-called cross ratio function (CRF), have been proposed to look at the association in more detail. This paper introduces a novel nonparametric estimator for the CRF applicable for univariate right-censored data, relying on Bernstein polynomials to obtain a smooth estimate of the bivariate survival copula, its partial derivatives, and the copula density. The proposed estimator's finite-sample performance is evaluated through an elaborate simulation study and applied to real-life data, highlighting its practical utility and setting the stage for future research on local association in survival analysis.