A simple adaptive estimator of the integrated square of a density

TL;DR

This paper introduces a simple, adaptive kernel-based estimator for the integrated square of a density, which is efficient for smooth densities and rate-optimal for less smooth ones, without prior smoothness knowledge.

Contribution

It proposes a data-driven bandwidth selection rule that makes the estimator both rate-adaptive and asymptotically efficient depending on the density's smoothness.

Findings

Estimator is asymptotically efficient for b1 > 1/4.

Estimator is rate-optimal for b1 a0a0 1/4.

Bandwidth selection rule is data-driven and does not require prior knowledge of b1.

Abstract

Given an i.i.d. sample $X_1,...,X_n$ with common bounded density $f_0$ belonging to a Sobolev space of order $\alpha$ over the real line, estimation of the quadratic functional $\int_{\mathbb{R}}f_0^2(x) \mathrm{d}x$ is considered. It is shown that the simplest kernel-based plug-in estimator \[\frac{2}{n(n-1)h_n}\sum_{1\leq i<j\leq n}K\biggl(\frac{X_i-X_j}{h_n}\biggr)\] is asymptotically efficient if $\alpha>1/4$ and rate-optimal if $\alpha\le1/4$. A data-driven rule to choose the bandwidth $h_n$ is then proposed, which does not depend on prior knowledge of $\alpha$, so that the corresponding estimator is rate-adaptive for $\alpha \leq1/4$ and asymptotically efficient if $\alpha>1/4$.