A simple adaptive estimator of the integrated square of a density

A simple adaptive estimator of the integrated square of a density

Evarist Giné and Richard Nickl

Source: Bernoulli Volume 14, Number 1 (2008), 47-61.

Abstract

Given an i.i.d. sample X₁, …, X_n with common bounded density f₀ belonging to a Sobolev space of order α over the real line, estimation of the quadratic functional ∫_ℝf₀²(x) dx is considered. It is shown that the simplest kernel-based plug-in estimator

$\frac{2}{n(n-1)h_{n}}\sum_{1\leq i\textless j\leq n}K\biggl(\frac {X_{i}-X_{j}}{h_{n}}\biggr)$

is asymptotically efficient if α>1/4 and rate-optimal if α≤1/4. A data-driven rule to choose the bandwidth h_n is then proposed, which does not depend on prior knowledge of α, so that the corresponding estimator is rate-adaptive for α≤1/4 and asymptotically efficient if α>1/4.

Keywords: adaptive estimation; kernel density estimator; quadratic functional

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.bj/1202492784
Digital Object Identifier: doi:10.3150/07-BEJ110

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