Large sample behavior of the least trimmed squares estimator

TL;DR

This paper introduces and studies the population version of the least trimmed squares (LTS) estimator, establishing its fundamental properties and large sample behavior to enhance understanding of its theoretical foundations.

Contribution

It is the first to define and analyze the population version of LTS, providing new insights into its properties and asymptotic behavior.

Findings

Established novel properties of the LTS objective function

Proved strong consistency using a generalized Glivenko-Cantelli Theorem

Derived asymptotic normality through differentiability and stochastic equicontinuity

Abstract

The least trimmed squares (LTS) estimator is popular in location, regression, machine learning, and AI literature. Despite the empirical version of least trimmed squares (LTS) being repeatedly studied in the literature, the population version of the LTS has never been introduced and studied. The lack of the population version hinders the study of the large sample properties of the LTS utilizing the empirical process theory. Novel properties of the objective function in both empirical and population settings of the LTS and other properties are established for the first time in this article. The primary properties of the objective function facilitate the establishment of other original results, including the influence function and Fisher consistency. The strong consistency is established with the help of a generalized Glivenko-Cantelli Theorem over a class of functions for the first time. Differentiability and stochastic equicontinuity promote the establishment of asymptotic normality with a concise and novel approach.