Asymptotics and Optimal Bandwidth Selection for Nonparametric Estimation of Density Level Sets

Bandwidth selection is crucial in the kernel estimation of density level sets. Risk based on the symmetric difference between the estimated and true level sets is usually used to measure their proximity. In this paper we provide an asymptotic $L_1$ approximation to this risk, propose a corresponding $L_2$ type of risk, and use the new risk to develop an optimal bandwidth selection rule for nonparametric level set estimation of $d$-dimensional density functions ($d\geq 1$). As an important ingredient of the selection rule, integrations over the boundary of the level sets with respect to the Hausdorff measure are studied and the convergence rate of their estimation is derived.