On the asymptotic properties of a simple estimate of the Mode

Abraham, Christophe, Biau, Gérard, and Cadre, Benoît. "On the asymptotic properties of a simple estimate of the Mode." ESAIM: Probability and Statistics 8 (2004): 1-11. <http://eudml.org/doc/245948>.

@article{Abraham2004,
abstract = {We consider an estimate of the mode $\theta $ of a multivariate probability density $f$ with support in $\mathbb \{R\}^d$ using a kernel estimate $f_n$ drawn from a sample $X_1, \hdots , X_n$. The estimate $\theta _n$ is defined as any $x$ in $\lbrace X_1, \hdots , X_n\rbrace $ such that $f_n(x)=\max _\{i=1, \hdots ,n\} f_n(X_i)$. It is shown that $\theta _n$ behaves asymptotically as any maximizer $\{\hat\{\theta \}\}_n$ of $f_n$. More precisely, we prove that for any sequence $(r_n)_\{n\ge 1\}$ of positive real numbers such that $r_n\rightarrow \infty $ and $r_n^d\log n/n\rightarrow 0$, one has $r_n\,\Vert \theta _n-\{\hat\{\theta \}\}_n\Vert \rightarrow 0$ in probability. The asymptotic normality of $\theta _n$ follows without further work.},
author = {Abraham, Christophe, Biau, Gérard, Cadre, Benoît},
journal = {ESAIM: Probability and Statistics},
keywords = {multivariate probability density; mode; kernel estimate; central limit theorem},
language = {eng},
pages = {1-11},
publisher = {EDP-Sciences},
title = {On the asymptotic properties of a simple estimate of the Mode},
url = {http://eudml.org/doc/245948},
volume = {8},
year = {2004},
}

TY - JOUR
AU - Abraham, Christophe
AU - Biau, Gérard
AU - Cadre, Benoît
TI - On the asymptotic properties of a simple estimate of the Mode
JO - ESAIM: Probability and Statistics
PY - 2004
PB - EDP-Sciences
VL - 8
SP - 1
EP - 11
AB - We consider an estimate of the mode $\theta $ of a multivariate probability density $f$ with support in $\mathbb {R}^d$ using a kernel estimate $f_n$ drawn from a sample $X_1, \hdots , X_n$. The estimate $\theta _n$ is defined as any $x$ in $\lbrace X_1, \hdots , X_n\rbrace $ such that $f_n(x)=\max _{i=1, \hdots ,n} f_n(X_i)$. It is shown that $\theta _n$ behaves asymptotically as any maximizer ${\hat{\theta }}_n$ of $f_n$. More precisely, we prove that for any sequence $(r_n)_{n\ge 1}$ of positive real numbers such that $r_n\rightarrow \infty $ and $r_n^d\log n/n\rightarrow 0$, one has $r_n\,\Vert \theta _n-{\hat{\theta }}_n\Vert \rightarrow 0$ in probability. The asymptotic normality of $\theta _n$ follows without further work.
LA - eng
KW - multivariate probability density; mode; kernel estimate; central limit theorem
UR - http://eudml.org/doc/245948
ER -