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On the asymptotic properties of a simple estimate of the Mode

Christophe AbrahamGérard BiauBenoît Cadre — 2004

ESAIM: Probability and Statistics

We consider an estimate of the mode θ of a multivariate probability density f with support in d using a kernel estimate f n drawn from a sample X 1 , , X n . The estimate θ n is defined as any x in { X 1 , , X n } such that f n ( x ) = max i = 1 , , n f n ( X i ) . It is shown that θ n behaves asymptotically as any maximizer θ ^ n of f n . More precisely, we prove that for any sequence ( r n ) n 1 of positive real numbers such that r n and r n d log n / n 0 , one has r n θ n - θ ^ n 0 in probability. The asymptotic normality of θ n follows without further work.

A graph-based estimator of the number of clusters

Gérard BiauBenoît CadreBruno Pelletier — 2007

ESAIM: Probability and Statistics

Assessing the number of clusters of a statistical population is one of the essential issues of unsupervised learning. Given independent observations drawn from an unknown multivariate probability density , we propose a new approach to estimate the number of connected components, or clusters, of the -level set ( t ) = { x : f ( x ) t } . The basic idea is to form a rough skeleton of the set ( t ) using any preliminary estimator of , and to count the number of connected components of the resulting graph. Under mild analytic...

On the asymptotic properties of a simple estimate of the Mode

Christophe AbrahamGérard BiauBenoît Cadre — 2010

ESAIM: Probability and Statistics

We consider an estimate of the mode of a multivariate probability density with support in d using a kernel estimate drawn from a sample . The estimate is defined as any in {} such that f n ( x ) = max i = 1 , , n f n ( X i ) . It is shown that behaves asymptotically as any maximizer θ ^ n of . More precisely, we prove that for any sequence ( r n ) n 1 of positive real numbers such that r n and r n d log n / n 0 , one has r n θ n - θ ^ n 0 in probability. The asymptotic normality of follows without further work.

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