A central limit theorem for the Hellinger loss of Grenander type estimators

TL;DR

This paper establishes a central limit theorem for the Hellinger loss of Grenander type estimators in monotone function estimation, providing new theoretical insights into their asymptotic behavior under certain models.

Contribution

It introduces a CLT for the Hellinger loss of Grenander estimators, extending previous results to a broader class of models and showing the variance independence in density estimation.

Findings

Central limit theorem for Hellinger loss established

Limiting variance is independent of the true function in density estimation

Results apply to models satisfying Durot (2007) setup

Abstract

We consider Grenander type estimators for a monotone function $\lambda:[0,1]\to\mathbb{R}$, obtained as the slope of a concave (convex) estimate of the primitive of $\lambda$. Our main result is a central limit theorem for the Hellinger loss, which applies to statistical models that satisfy the setup in Durot (2007). This includes estimation of a monotone density, for which the limiting variance of the Hellinger loss turns out to be independent of $\lambda$.