Asymptotics and Optimal Bandwidth for Nonparametric Estimation of Density Level Sets

Bandwidth selection is crucial in the kernel estimation of density level sets. A 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 to derive an optimal bandwidth for nonparametric level set estimation of $d$-dimensional density functions ($d\geq 1$). A direct plug-in bandwidth selector is developed for kernel density level set estimation and its efficacy is verified in numerical studies.