Kernel estimation of density level sets

Abstract. 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. Key-words: Kernel estimate, Density level sets, Hausdorff measure. 2000 Mathematics Subject Classification: 62H12, 62H30. Introduction. Stimulated by a growing demand from applied results, the theory of density level sets estimation has developed significantly over the last few years. One of the most important application of density level sets estimation is in unsupervised cluster analysis (see for instance, Hartigan [1], Cuevas, Febrero and Fraiman [2,3]). Here, one tries to break a complex dat