Optimal rates for plug-in estimators of density level sets
Philippe Rigollet and Régis Vert
Source: Bernoulli Volume 15, Number 4
(2009), 1154-1178.
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 λ. More precisely, it is assumed that the density (i) is smooth in a neighborhood of λ and (ii) has γ-exponent at level λ. 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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Keywords: density level sets; kernel density estimators; minimax lower bounds; plug-in estimators; rates of convergence
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Permanent link to this document: http://projecteuclid.org/euclid.bj/1262962230
Digital Object Identifier: doi:10.3150/09-BEJ184
Zentralblatt MATH identifier: 1200.62034
Mathematical Reviews number (MathSciNet): MR2597587
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