Estimation of the weighted integrated square error of the Grenander estimator by the Kolmogorov-Smirnov statistic

TL;DR

This paper establishes a probabilistic bound on the weighted integrated square error of the Grenander estimator using the Kolmogorov-Smirnov statistic, applicable to nonincreasing densities on [0,1] without smoothness assumptions.

Contribution

It provides a novel inequality linking the error of the Grenander estimator to the Kolmogorov-Smirnov statistic for general nonincreasing densities.

Findings

Bound on the weighted integrated square error with probability one

Implications for non-smooth, nonincreasing density estimation

Connection between estimation error and Kolmogorov-Smirnov statistic

Abstract

We consider in this paper the Grenander estimator of unbounded, in general, nonincreasing densities on the interval [0; 1] without any smoothness assumptions. For fixed number n of i.i.d. random vari- ables X1;X2; : : : ;Xn with values in [0; 1] and the nonincreasing den- sity function f(x), 0 < x < 1, we prove an inequality bounding the weighted integrated square error of the Grenander estimator with probability one by the classical Kolmogorov-Smirnov statistic. Fur- ther, we consider some interesting implications of the latter inequality