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 $L^p$ approximation to this risk, where $p$ is characterized by the weight function in the risk. In particular the excess risk corresponds to an $L^2$ type of risk, and is adopted in an optimal bandwidth selection rule for nonparametric level set estimation of $d$-dimensional density functions ($d\geq 1$).