On the asymptotic properties of a simple estimate of the Mode

Abraham, Christophe; Biau, Gérard; Cadre, Benoît

doi:10.1051/ps:2003015

Abraham, Christophe ; Biau, Gérard ; Cadre, Benoît

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

Résumé

We consider an estimate of the mode $θ$ of a multivariate probability density $f$ with support in $ℝ^{d}$ using a kernel estimate $f_{n}$ drawn from a sample $X_{1}, \dots, X_{n}$ . The estimate $θ_{n}$ is defined as any $x$ in ${X_{1}, \dots, X_{n}}$ such that $f_{n} (x) = {max}_{i = 1, \dots, n} f_{n} (X_{i})$ . It is shown that $θ_{n}$ behaves asymptotically as any maximizer ${\hat{θ}}_{n}$ 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 $r_{n} ∥ θ_{n} - {\hat{θ}}_{n} ∥ \to 0$ in probability. The asymptotic normality of $θ_{n}$ follows without further work.

MR Zbl

DOI : 10.1051/ps:2003015

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

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     title = {On the asymptotic properties of a simple estimate of the {Mode}},
     journal = {ESAIM: Probability and Statistics},
     pages = {1--11},
     year = {2004},
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Abraham, Christophe; Biau, Gérard; Cadre, Benoît. On the asymptotic properties of a simple estimate of the Mode. ESAIM: Probability and Statistics, Tome 8 (2004), pp. 1-11. doi: 10.1051/ps:2003015

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