Asymptotics and optimal bandwidth selection for highest density region estimation

TL;DR

This paper develops a theoretical framework for kernel estimation of highest-density regions, deriving asymptotic risk approximations and an optimal bandwidth selection rule, supported by numerical validation.

Contribution

It introduces a uniform-in-bandwidth asymptotic approximation for HDR risk and proposes a new bandwidth selection method with strong asymptotic properties.

Findings

The asymptotic approximation accurately predicts HDR estimation risk.

The proposed bandwidth rule improves HDR estimation performance.

Numerical studies confirm the effectiveness of the new methodology.

Abstract

We study kernel estimation of highest-density regions (HDR). Our main contributions are two-fold. First, 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.