Estimation of the Global Mode of a Density: Minimaxity, Adaptation, and Computational Complexity

TL;DR

This paper investigates methods for estimating the global mode of a density, demonstrating minimax optimality with histogram-based estimators and proposing an adaptive multiscale approach that is computationally efficient.

Contribution

It introduces a minimax optimal histogram estimator for known decay rates and a minimax adaptive multiscale method for unknown decay rates, with proven linear time complexity.

Findings

Histogram estimator achieves minimax rate with proper bandwidth.

Multiscale adaptive estimator is minimax optimal for unknown decay.

Estimation procedures cannot be sublinear time and still achieve minimax rate.

Abstract

We consider the estimation of the global mode of a density under some decay rate condition around the global mode. We show that the maximum of a histogram, with proper choice of bandwidth, achieves the minimax rate that we establish for the setting that we consider. This is based on knowledge of the decay rate. Addressing the situation where the decay rate is unknown, we propose a multiscale variant consisting in the recursive refinement of a histogram, which is shown to be minimax adaptive. These methods run in linear time, and we prove in an appendix that this is best possible: There is no estimation procedure that runs in sublinear time that achieves the minimax rate.