Mixed-rates asymptotics

TL;DR

This paper introduces a new general method for analyzing the asymptotic behavior of estimators defined as optimizers of empirical criterion functions, especially in cases where standard methods are insufficient.

Contribution

It provides a novel approach based on decomposing criterion functions into components with different rescalings, applicable to various estimation problems.

Findings

Applicable to Lasso-type estimation

Effective for k-means clustering analysis

Handles partial linear models

Abstract

A general method is presented for deriving the limiting behavior of estimators that are defined as the values of parameters optimizing an empirical criterion function. The asymptotic behavior of such estimators is typically deduced from uniform limit theorems for rescaled and reparametrized criterion functions. The new method can handle cases where the standard approach does not yield the complete limiting behavior of the estimator. The asymptotic analysis depends on a decomposition of criterion functions into sums of components with different rescalings. The method is explained by examples from Lasso-type estimation, $k$-means clustering, Shorth estimation and partial linear models.