On the asymptotic behavior of the contaminated sample mean

TL;DR

This paper studies the asymptotic properties of the sample mean under a contaminated distribution model with inflated variance, analyzing its consistency and normality using advanced central limit theory.

Contribution

It provides conditions under which the contaminated sample mean remains a valid and asymptotically normal estimator for the true mean.

Findings

Sample mean can be weakly consistent under certain contamination conditions.

Asymptotic normality of the sample mean is established using approximate CLT.

Simulation confirms theoretical results.

Abstract

An observation of a cumulative distribution function $F$ with finite variance is said to be contaminated according to the inflated variance model if it has a large probability of coming from the original target distribution $F$, but a small probability of coming from a contaminating distribution that has the same mean and shape as $F$, though a larger variance. It is well known that in the presence of data contamination, the ordinary sample mean looses many of its good properties, making it preferable to use more robust estimators. From a didactical point of view, it is insightful to see to what extent an intuitive estimator such as the sample mean becomes less favorable in a contaminated setting. In this paper, we investigate under which conditions the sample mean, based on a finite number of independent observations of $F$ which are contaminated according to the inflated variance model, is a valid estimator for the mean of $F$. In particular, we examine to what extent this estimator is weakly consistent for the mean of $F$ and asymptotically normal. As classical central limit theory is generally inaccurate to cope with the asymptotic normality in this setting, we invoke more general approximate central limit theory as developed by Berckmoes, Lowen, and Van Casteren (2013). Our theoretical results are illustrated by a specific example and a simulation study.