A New Smooth Nonparametric Kernel Estimator of the Cross Ratio Function

Accepted - May 2025

Abstract

This research was inspired by the well-known cross ratio function (CRF) proposed by Clayton (1978), which is a local dependence measure describing the association between the components of a bivariate random vector. A new smooth, fully nonparametric and easy-to-implement kernel estimator of the CRF is introduced and studied. Its asymptotic distribution is derived and a result of independent interest is obtained. The outcome of a numerical study based on simulated data shows that the proposed estimator has, in terms of mean integrated squared error, favourable performance compared to the smooth Bernstein estimator recently constructed by Abrams et al. (2020). Furthermore, from a practical perspective, the proposed estimator is computationally much less demanding and we provide a plug-in method for choosing the smoothing parameters.