Bernoulli

  • Bernoulli
  • Volume 15, Number 4 (2009), 1154-1178.

Optimal rates for plug-in estimators of density level sets

Philippe Rigollet and 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 $λ$. 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.

Article information

Source
Bernoulli, Volume 15, Number 4 (2009), 1154-1178.

Dates
First available in Project Euclid: 8 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.bj/1262962230

Digital Object Identifier
doi:10.3150/09-BEJ184

Mathematical Reviews number (MathSciNet)
MR2597587

Zentralblatt MATH identifier
1200.62034

Keywords
density level sets kernel density estimators minimax lower bounds plug-in estimators rates of convergence

Citation

Rigollet, Philippe; Vert, Régis. Optimal rates for plug-in estimators of density level sets. Bernoulli 15 (2009), no. 4, 1154--1178. doi:10.3150/09-BEJ184. https://projecteuclid.org/euclid.bj/1262962230


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