Central limit theorems for the $L_p$-error of smooth isotonic estimators

TL;DR

This paper establishes central limit theorems for the $L_p$-error of smooth isotonic estimators, providing a theoretical foundation for their asymptotic behavior in various monotone function estimation problems.

Contribution

It introduces a general CLT for the $L_p$-error of smooth isotonic estimators, including smoothing of Grenander-type and kernel estimators, applicable to multiple monotone function estimations.

Findings

CLT for $L_p$-error of smooth isotonic estimators

Asymptotic normality results for various monotone estimations

Simulation study validating the theoretical results

Abstract

We investigate the asymptotic behavior of the $L_p$-distance between a monotone function on a compact interval and a smooth estimator of this function. Our main result is a central limit theorem for the $L_p$-error of smooth isotonic estimators obtained by smoothing a Grenander-type estimator or isotonizing the ordinary kernel estimator. As a preliminary result we establish a similar result for ordinary kernel estimators. Our results are obtained in a general setting, which includes estimation of a monotone density, regression function and hazard rate. We also perform a simulation study for testing monotonicity on the basis of the $L_2$-distance between the kernel estimator and the smoothed Grenander-type estimator.