On the asymptotic properties of a simple estimate of the Mode

Abraham, Christophe  ; Biau, Gérard  ; Cadre, Benoît 

ESAIM: Probability and Statistics, Tome 8 (2004), pp. 1-11

Résumé

We consider an estimate of the mode $\theta$ of a multivariate probability density $f$ with support in ${ℝ}^d$ using a kernel estimate $f_n$ drawn from a sample $X_1,\cdots ,X_n$. The estimate ${\theta }_n$ is defined as any $x$ in $\{X_1,\cdots ,X_n\}$ such that $f_n\left(x\right)={max}_{i=1,\cdots ,n}{f}_n\left(X_i\right)$. It is shown that ${\theta }_n$ behaves asymptotically as any maximizer ${\widehat{\theta }}_n$ of $f_n$. More precisely, we prove that for any sequence ${\left(r_n\right)}_{n\ge 1}$ of positive real numbers such that $r_n\to \infty$ and $r_n^{d}logn/n\to 0$, one has $r_n\parallel {\theta }_n-{\widehat{\theta }}_n\parallel \to 0$ in probability. The asymptotic normality of ${\theta }_n$ follows without further work.

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DOI : 10.1051/ps:2003015

Classification : 62G05
Keywords: multivariate probability density, mode, kernel estimate, central limit theorem