On Estimating $L_2^2$ Divergence

TL;DR

This paper provides a detailed theoretical analysis of a nonparametric estimator for the $L_2^2$ divergence between continuous distributions, including convergence rates, asymptotic normality, confidence intervals, and minimax optimality.

Contribution

It offers the first comprehensive theoretical characterization of a nonparametric $L_2^2$ divergence estimator, including convergence, normality, and optimality results.

Findings

Estimator is $ oot{n}$-consistent for smooth densities.

Estimator is asymptotically normal with confidence intervals.

Estimator is minimax optimal.

Abstract

We give a comprehensive theoretical characterization of a nonparametric estimator for the $L_2^2$ divergence between two continuous distributions. We first bound the rate of convergence of our estimator, showing that it is $\sqrt{n}$-consistent provided the densities are sufficiently smooth. In this smooth regime, we then show that our estimator is asymptotically normal, construct asymptotic confidence intervals, and establish a Berry-Ess\'{e}en style inequality characterizing the rate of convergence to normality. We also show that this estimator is minimax optimal.