Bernstein-based estimation of the cross-ratio function
My role: Co-investigator
Project period: 2022–2025
FWO Junior Fundamental Research Project · G011022N
Research summary
The cross ratio function, originally introduced by Clayton (1978), describes local dependence that a global association measure can conceal. This line of research uses Bernstein polynomials to obtain a smooth, nonparametric estimator of the cross ratio function without imposing a fixed parametric form for the dependence. More specifically, to date nonparametric estimators have been developed for bivariate time-to-event data, including data subject to univariate right censoring, and for uncensored continuous data.
The methodological foundation is found in Abrams et al. (2020). In this paper, we construct a Bernstein-based estimator for the cross ratio function through estimating the conditional hazard functions as numerator and denominator of the cross ratio function, and we establish asymptotic normality of the estimator, and evaluate its finite-sample performance through simulations. Finally, a data application to household food expenditure and income illustrates the proposed estimator. In Abrams et al. (2024), we propose an alternative Bernstein-based plug-in estimator based on the estimation of the joint survival function and its derivatives. We establish asymptotic normality of the estimator and assess the performance of the estimator using simulations and based on real-life hospital data. Both papers focus on the analysis of uncensored bivariate time-to-event data.
Sercik et al. (2026) extend the aforementioned approach to data subject to univariate right censoring. Bernstein estimates of the survival copula, its partial derivatives and its density provide a smooth estimate of local association across the bivariate domain, with particular attention to regions with sufficient uncensored information. Simulation experiments and a real-data application assess its finite-sample performance.
Related publications
- Abrams et al. (2020) — Nonparametric estimation of the cross ratio function. Annals of the Institute of Statistical Mathematics, 72, 771–801.
- Abrams et al. (2024) — Bernstein-based estimation of the cross ratio function. Statistics, 58(1), 230–246.
- Sercik et al. (2026) — Bernstein-based nonparametric estimation of the cross ratio function under univariate right censoring. Dependence Modeling, 14(1).