Large and moderate deviations principles for recursive kernel estimators of a multivariate density and its partial derivatives

TL;DR

This paper establishes large and moderate deviations principles for recursive kernel estimators of multivariate densities and their derivatives, highlighting quadratic behavior in derivatives across deviation scales.

Contribution

It introduces deviation principles for derivatives of recursive kernel estimators, revealing quadratic behavior unlike the density estimator, for both pointwise and uniform deviations.

Findings

Derivatives estimators show quadratic behavior in deviations

Results apply to both pointwise and uniform deviations

Large and moderate deviations principles are established

Abstract

In this paper we prove large and moderate deviations principles for the recursive kernel estimator of a probability density function and its partial derivatives. Unlike the density estimator, the derivatives estimators exhibit a quadratic behavior not only for the moderate deviations scale but also for the large deviations one. We provide results both for the pointwise and the uniform deviations.