Estimation of density level sets with a given probability content

摘要

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 {f≥ t~((p))} of f corresponding to probability content p, that is, such that the probability that X belongs to {f≥ t~((p)) is equal to p. Based on an i.i.d. random sample X_l,...,X_n drawn from f, we define the plug-in level set estimate {f_n≥t_n~((p))}, where t_n~((p)) is a random threshold depending on the sample and f_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.