## Bernoulli

### A simple adaptive estimator of the integrated square of a density

#### Abstract

Given an i.i.d. sample $X_1, …, X_n$ with common bounded density $f_0$ belonging to a Sobolev space of order $α$ over the real line, estimation of the quadratic functional $∫_ℝf_0^2(x) \mathrm{d}x$ 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 $α>1/4$ 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 $α>1/4$.

#### Article information

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

Dates
First available in Project Euclid: 8 February 2008

https://projecteuclid.org/euclid.bj/1202492784

Digital Object Identifier
doi:10.3150/07-BEJ110

Mathematical Reviews number (MathSciNet)
MR2401653

Zentralblatt MATH identifier
1155.62025

#### Citation

Giné, Evarist; Nickl, Richard. A simple adaptive estimator of the integrated square of a density. Bernoulli 14 (2008), no. 1, 47--61. doi:10.3150/07-BEJ110. https://projecteuclid.org/euclid.bj/1202492784

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