Estimation of density level sets with a given probability content

Abstract

Given a random vector X valued in ℝ ^d with density f and an arbitrary probability number p∈(0; 1), we consider the estimation of the upper level set<texlscub>f≥t ^(p)</texlscub>of f corresponding to probability content p, that is, such that the probability that X belongs to<texlscub>f≥t ^(p)</texlscub>is equal to p. Based on an i.i.d. random sample X ₁, …, X _n drawn from f, we define the plug-in level set estimate , where is a random threshold depending on the sample and [fcirc] _n is a nonparametric kernel density estimate based on the same sample. We establish the exact convergence rate of the Lebesgue measure of the symmetric difference between the estimated and actual level sets.

Keywords:

• kernel estimate
• density level sets
• nonparametric statistics

AMS 2000 Classifications:

• 62H30
• 62H12

Acknowledgements

This work was partially supported by a grant from the French National Research Agency (ANR 09-BLAN-0051-01). We thank two anonymous referees and the Associate Editor for valuable comments and insightful suggestions.