Kernel estimation of density level sets

Let f be a multivariate density and fn be a kernel estimate of f drawn from the n-sample X1,…,Xn 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 {fn⩾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.