Asymptotics and optimal bandwidth selection for highest density region estimation

We study kernel estimation of highest density regions (HDR). Our main contributions are twofold. Firstly, we derive a uniform-in-bandwidth asymptotic approximation to a risk that is appropriate for HDR estimation. This approximation is then used to derive a bandwidth selection rule for HDR estimation possessing attractive asymptotic properties. We also present the results of numerical studies that illustrate the benefits of our theory and methodology.