Title: Nonparametric estimation of the cross ratio function
Authors: Noel Veraverbeke - Hasselt University (Belgium) [presenting]
Abstract: For a pair $(T_1,T_2)$ of absolutely continuous random variables, the cross ratio function is defined as the ratio of the conditional hazard rate functions of $T1$, given $T_2 = t_2$ and $T_2 > t_2$ respectively. Independence between $T_1$ and $T_2$ corresponds to cross ratio equal to 1 and positive association corresponds to cross ratio > 1. Nowadays the cross ratio function is a commonly used measure to describe local dependence between two correlated random variables. Being a ratio of conditional hazard functions, the cross ratio can be written in terms of the survival copula of $T_1$ and $T_2$ and its partial derivatives. Using Bernstein estimators for the survival copula and its derivatives, we obtain Bernstein based estimators for the conditional hazards and a nonparametric estimator for the cross ratio function. The reason for using a Bernstein copula-based estimator for the cross ratio function is motivated from earlier results showing good bias and variance properties. The asymptotic distributional behavior of the new estimator is established. We also consider a number of simulations to study the finite sample performance for copulas with different types of local dependence. A real data set on asthma attacks in children is used to investigate the local dependence between event times in the placebo and treated groups.