On the asymptotic properties of a simple estimate of the Mode

@article{Abraham2010,
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 fn 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 fn. 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. },
author = {Abraham, Christophe, Biau, Gérard, Cadre, Benoît},
journal = {ESAIM: Probability and Statistics},
keywords = {Multivariate probability density; mode; kernel estimate; central limit theorem.; multivariate probability density; central limit theorem},
language = {eng},
month = {3},
pages = {1-11},
publisher = {EDP Sciences},
title = {On the asymptotic properties of a simple estimate of the Mode},
url = {http://eudml.org/doc/104318},
volume = {8},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 1
EP - 11
AB - We consider an estimate of the mode θ of a multivariate probability density f with support in $\mathbb R^d$ using a kernel estimate fn 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 fn. 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.
LA - eng
KW - Multivariate probability density; mode; kernel estimate; central limit theorem.; multivariate probability density; central limit theorem
UR - http://eudml.org/doc/104318
ER -