Generalised local polynomial estimators of smooth functionals of a distribution function with nonnegative support

Abstract

This paper introduces generalised smooth asymmetric kernel estimators for smooth functionals with non-negative support. More precisely, for x ∈ [ 0 , ∞ ) , and for a functional, Φ ( x , F ) of the distribution function F, we develop estimators of the functional Φ and its derivatives. The proposed estimator can be seen as the solution to a minimisation problem in the polynomial space L 2 ( q ) , where q is an asymmetric density function. The framework presented here covers several classical nonparametric functional estimators and is linked with estimation using hierarchical kernels. We establish the asymptotic properties of the proposed estimators in the general framework. Furthermore, special attention is paid to comparing the asymptotic mean integrated square error (AMISE) of the proposed estimator with that of other classical symmetric/asymmetric density estimators. Additionally, a comparison of finite sample behaviour is conducted for both density estimation and hazard rate estimation via simulation and real data application.