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

Abstract

In the Supplementary Material below, we present additional results discussed in the main text of this manuscript. In Section , we show the results of the Clayton copula for other configurations. Furthermore, in Section , we provide the results of the independence, Gumbel and Frank copulas, respectively in Section , Section and Section. For these copulas, we give the results for n = 400, π C = 50% and τ K = 0.5. Finally, in Section , we show some additional results regarding the data application. S1 Additional Results of Clayton Copula Additional to the results for the Clayton copula given in the main text, we also provide the results optimized for other settings. In particular, in the main text, we have shown the results for n = 400, π C = 50% and τ K = 0.5. We now show the results for n = 200, π C = 50%, τ K = 0.5, n = 800, π C = 50%, τ K = 0.5, n = 400, π C = 30%, τ K = 0.5, n = 400, π C = 70%, τ K = 0.5, n = 400, π C = 50%, τ K = 0.2 and n = 400, π C = 50, τ K = 0.8. These results are given, respectively, in Figure S1, Figure S2, Figure S3, Figure S4, Figure S5 and Figure S6.

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 rightcensored 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.