Abstract
Given a random vector X valued in R^d with density f and an arbitrary probability number p in (0; 1), we consider the estimation of the upper level set of f corresponding to probability content p, that is, such that the probability that X belongs to it 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 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.