Recursive estimators of integrated squared density derivatives

摘要

Hall and Marron (1987) introduced kernel estimators of integrals of various squared derivatives of a probability density. The aim of this paper is the study of recursive versions of their estimators. Rates of convergence in mean squared error (MSE) are calculated. Similarly to the estimators of Hall and Marron (1987), the recursive estimators may achieve the parametric rate n(-1); the striking fact is that their MSE are then equivalent to those of their nonrecursive versions, whereas the recursive nonparametric estimators are known for usually having larger MSE than their nonrecursive version. We also provide recursive estimators of the optimal bandwidth in the framework of density estimation. (C) 2019 Elsevier B.V. All rights reserved.