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Preprints, Working Papers, ... Year : 2008

Fast rates for plug-in estimators of density level sets

Régis Vert
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Abstract

In the context of density level set estimation, we study the convergence of general plug-in methods under two main assumptions on the density for a given level $\lambda$. More precisely, it is assumed that the density (i) is smooth in a neighborhood of $\lambda$ and (ii) has $\gamma$-exponent at level $\lambda$. Condition (i) ensures that the density can be estimated at a standard nonparametric rate and condition (ii) is similar to Tsybakov's margin assumption which is stated for the classification framework. Under these assumptions, we derive optimal rates of convergence for plug-in estimators. Explicit convergence rates are given for plug-in estimators based on kernel density estimators when the underlying measure is the Lebesgue measure. Lower bounds proving optimality of the rates in a minimax sense when the density is Hölder smooth are also provided.
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Dates and versions

hal-00114180 , version 1 (15-11-2006)
hal-00114180 , version 2 (24-01-2007)
hal-00114180 , version 3 (19-02-2008)
hal-00114180 , version 4 (16-10-2008)

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Philippe Rigollet, Régis Vert. Fast rates for plug-in estimators of density level sets. 2008. ⟨hal-00114180v4⟩
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