On the asymptotic properties of a simple estimate of the Mode

Christophe Abraham; Gérard Biau; Benoît Cadre

doi:10.1051/ps:2003015

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Volume 8 (August 2004)

ESAIM: PS, 8 (2004) 1-11

Abstract

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Issue		ESAIM: PS Volume 8, August 2004


Page(s)		1 - 11
DOI		https://doi.org/10.1051/ps:2003015
Published online		15 September 2004

ESAIM: P&S, January 2004, Vol. 8, p. 1-11

On the asymptotic properties of a simple estimate of the Mode

Christophe Abraham¹, Gérard Biau² and Benoît Cadre³

¹ ENSAM-INRA, UMR Biométrie et Analyse des Systèmes, 2 place Pierre Viala, 34060 Montpellier Cedex 1, France; abraham@helios.ensam.inra.fr.
² Laboratoire de Statistique Théorique et Appliquée, Université Pierre et Marie Curie – Paris VI, Boîte 158, 175 rue du Chevaleret, 75013 Paris, France; biau@ccr.jussieu.fr.
³ Laboratoire de Probabilités et Statistique, Université Montpellier II, Cc. 051, place Eugène Bataillon, 34095 Montpellier Cedex 5, France; cadre@stat.math.univ-montp2.fr.

Received: 15 May 2002
Revised: 18 October 2002

Abstract

We consider an estimate of the mode θ of a multivariate probability density f with support in $\mathbb R^d$ using a kernel estimate f_n drawn from a sample X₁,...,X_n. The estimate θ_n is defined as any x in {X₁,...,X_n} such that $f_n(x)=\max_{i=1, \hdots,n} f_n(X_i)$ . It is shown that θ_n behaves asymptotically as any maximizer ${\hat{\theta}}_n$ of f_n. More precisely, we prove that for any sequence $(r_n)_{n\geq 1}$ of positive real numbers such that $r_n\to\infty$ and $r_n^d\log n/n\to 0$ , one has $r_n\,\|\theta_n-{\hat{\theta}}_n\| \to 0$ in probability. The asymptotic normality of θ_n follows without further work.

Mathematics Subject Classification: 62G05

Key words: Multivariate probability density / mode / kernel estimate / central limit theorem.

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