Adaptation to lowest density regions with application to support recovery

Piotr Kokoschka (Colorado State University)
Wednesday, June 18, 2014 - 10:00am
Mohrenstraße 39, Erhard-Schmidt-Hörsaal

A new scheme for locally adaptive bandwidth selection is proposed which sensitively shrinks the bandwidth of a kernel estimator at lowest density regions such as the support boundary which are unknown to the statistician. In case of a Hölder continuous density, this locally minimax-optimal bandwidth is shown to be smaller than the usual rate, even in case of homogeneous smoothness. Besides the classical minimax risk bounds at some fixed point, new pointwise risk bounds along a shrinking neighborhood of lowest density regions are derived, which demonstrate the superiority of the new estimator as compared to classical adaptive estimators. Our bounds are complemented by a local minimax lower bound. This lower bound splits into three regimes depending on the value of the density. The new estimator adapts to the first two regimes, and it is shown that simultaneous adaptation to the fastest regime is not possible in principal. The results are fully non-asymptotic. Consequences on plug-in rules for support recovery based on the new estimator are worked out in detail. In contrast to those built with classical density estimators, the plug-in rules based on the new construction are minimax-optimal, up to some logarithmic factor. As a by-product, we demonstrate that the rates on support estimation obtained in Cuevas and Fraiman (1997, Ann. Statist.) are always suboptimal in case of Hölder continuous densities.
This is a joint work with Tim Patschkowski.