Joint behaviour of semirecursive kernel estimators of the location and of the size of the mode of a probability density function

TL;DR

This paper investigates the joint convergence properties of semirecursive kernel estimators for a density mode's location and size, highlighting their advantages in computational efficiency and confidence region construction.

Contribution

It provides the first analysis of the joint convergence rates of semirecursive estimators for both mode location and size, emphasizing their practical benefits.

Findings

Semirecursive estimators have favorable convergence rates.

Estimating mode size helps assess location estimation relevance.

Semirecursive estimators outperform nonrecursive ones in confidence region construction.

Abstract

Let $\theta$ and $\mu$ denote the location and the size of the mode of a probability density. We study the joint convergence rates of semirecursive kernel estimators of $\theta$ and $\mu$. We show how the estimation of the size of the mode allows to measure the relevance of the estimation of its location. We also enlighten that, beyond their computational advantage on nonrecursive estimators, the semirecursive estimators are preferable to use for the construction on confidence regions.