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On the asymptotic properties of a simple estimate of the Mode

Published online by Cambridge University Press:  15 September 2004

Christophe Abraham
Affiliation:
ENSAM-INRA, UMR Biométrie et Analyse des Systèmes, 2 place Pierre Viala, 34060 Montpellier Cedex 1, France; [email protected].
Gérard Biau
Affiliation:
Laboratoire de Statistique Théorique et Appliquée, Université Pierre et Marie Curie – Paris VI, Boîte 158, 175 rue du Chevaleret, 75013 Paris, France; [email protected].
Benoît Cadre
Affiliation:
Laboratoire de Probabilités et Statistique, Université Montpellier II, Cc. 051, place Eugène Bataillon, 34095 Montpellier Cedex 5, France; [email protected].
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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,...,Xn . The estimate θn is defined as any x in {X1,...,Xn } such that $f_n(x)=\max_{i=1, \hdots,n} f_n(X_i)$ . It is shown that θn 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 θn follows without further work.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2004

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References

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