Moderate deviations for the $L_1$-norm of kernel density estimators

TL;DR

This paper investigates the rate of normal approximation for the L1-norm of kernel density estimators, especially for densities with power-type singularities and zeros, providing estimates for zones of moderate deviations.

Contribution

It extends existing results by estimating the size of zones of moderate deviations for densities with power-type singularities and zeros.

Findings

Derived estimates for the size of zones of moderate deviations.

Analyzed densities with power-type singularities and zeros.

Extended normal approximation results to more complex density behaviors.

Abstract

The rate of normal approximation for the integral norm of kernel density estimators is investigated in the case of densities with power-type singularities. The quantities from the formulations of published results by the author are estimated. By assumption, the density tends to zero as a power-type function when the argument tends to infinity. Moreover, the density may have a finite number of power-type zeroes and of points with power-type tending to infinity. For such densities the size of zones of moderate deviations are found.