Kernel estimation of density level sets

Let f be a multivariate density and f n be a kernel estimate of f drawn from the n-sample X 1 , … , X n of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the t-level set { f ⩾ t } and its plug-in estimator { f n ⩾ t } . As a corollary, we obtain the exact rate of convergence of a plug-in-type estimate of the density level set corresponding to a fixed probability for the law induced by f.