Monotone representation and measurability of generalized $\psi$-estimators

TL;DR

This paper studies the properties of generalized $ ext{ extpsi}$-estimators, focusing on their monotone representation and measurability, and establishes conditions linking estimator measurability to the underlying function $ ext{ extpsi}$.

Contribution

It provides a new representation of generalized $ ext{ extpsi}$-estimators and links their measurability to the measurability of the defining function $ ext{ extpsi}$ under certain conditions.

Findings

Constructed the estimator using a decreasing function $ ext{ extpsi}$.

Linked the measurability of the estimator to the measurability of $ ext{ extpsi}$.

Bridged nonconvex and convex optimization problems through this representation.

Abstract

We investigate the monotone representation and measurability of generalized $\psi$-estimators introduced by the authors in 2022. Our first main result, applying the unique existence of a generalized $\psi$-estimator, allows us to construct this estimator in terms of a function $\psi$, which is decreasing in its second variable. We then interpret this result as a bridge from a nonconvex optimization problem to a convex one. Further, supposing that the underlying measurable space (sample space) has a measurable diagonal and some additional assumptions on $\psi$, we show that the measurability of a generalized $\psi$-estimator is equivalent to the measurability of the corresponding function $\psi$ in its first variable.