A law of the iterated logarithm for Grenander's estimator

TL;DR

This paper establishes a law of the iterated logarithm for Grenander's estimator of a monotone decreasing density, detailing the almost sure asymptotic behavior of the estimator at a point under certain smoothness conditions.

Contribution

It proves a new law of the iterated logarithm for Grenander's estimator, connecting it with Strassen's limit set and local empirical process laws, which was previously unestablished.

Findings

The limsup of the scaled estimator converges almost surely to a specific constant.

The result links the estimator's fluctuations to properties of the Strassen limit set.

Provides a precise asymptotic characterization of the estimator's behavior at a point.

Abstract

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If $f(t_0) > 0$, $f'(t_0) < 0$, and $f'$ is continuous in a neighborhood of $t_0$, then \begin{eqnarray*} \limsup_{n\rightarrow \infty} \left ( \frac{n}{2\log \log n} \right )^{1/3} ( \widehat{f}_n (t_0 ) - f(t_0) ) = \left| f(t_0) f'(t_0)/2 \right|^{1/3} 2M \end{eqnarray*} almost surely where $ M \equiv \sup_{g \in {\cal G}} T_g = (3/4)^{1/3}$ and $ T_g \equiv \mbox{argmax}_u \{ g(u) - u^2 \} $; here ${\cal G}$ is the two-sided Strassen limit set on $R$. The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom's switching relation, and properties of Strassen's limit set analogous to distributional properties of Brownian motion.