The limit distribution of the $L_{\infty}$-error of Grenander-type estimators

TL;DR

This paper establishes the asymptotic Gumbel distribution for the supremum norm error of Grenander-type estimators of nonincreasing functions, revealing a convergence rate of (n/log n)^{-1/3}.

Contribution

It derives the limit distribution and convergence rate of the supremum error for Grenander-type estimators under standard regularity conditions.

Findings

Convergence rate of (n/log n)^{-1/3} for the supremum error.

Limiting distribution of the error is Gumbel.

Provides theoretical foundation for the behavior of Grenander estimators.

Abstract

Let $f$ be a nonincreasing function defined on $[0,1]$. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between $f$ and its Grenander-type estimator on sub-intervals of $[0,1]$. The rate of convergence is found to be of order $(n/\log n)^{-1/3}$ and the limiting distribution to be Gumbel.