Technical report: Adaptivity and optimality of the monotone least squares estimator for four different models

TL;DR

This paper investigates the adaptivity and optimality of the monotone least squares estimator for estimating monotone functions, demonstrating its fully adaptive nature and optimality across various models and conditions.

Contribution

It establishes the estimator's adaptivity, optimality, and provides conditions for its limiting distribution in different models.

Findings

Estimator is fully adaptive with respect to the underlying function.

The rate of convergence is optimal among all monotone functions.

Conditions are provided for the limiting distribution of the estimator.

Abstract

In this paper we will consider the estimation of a monotone regression (or density) function in a fixed point by the least squares (Grenander) estimator. We will show that this estimator is fully adaptive, in the sense that the attained rate is given by a functional relation using the underlying function $f_0$, and not by some smoothness parameter, and that this rate is optimal when considering the class of all monotone functions, in the sense that there exists a sequence of alternative monotone functions $f_1$, such that no other estimator can attain a better rate for both $f_0$ and $f_1$. We also show that under mild conditions the estimator attains the same rate in $L^q$ sense, and we give general conditions for which we can calculate a (non-standard) limiting distribution for the estimator.