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Mathematical Methods of Statistics

, Volume 27, Issue 4, pp 312–323 | Cite as

On the Asymptotic Behavior of the Contaminated Sample Mean

  • B. BerckmoesEmail author
  • G. Molenberghs
Article
  • 3 Downloads

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. 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 copewith the asymptotic normality in this setting, we invokemore general approximate central limit theory as developed in [3]. Our theoretical results are illustrated by a specific example and a simulation study.

Keywords

approximate central limit theory asymptotic normality consistency contaminated data Kolmogorov distance Lindeberg index sample mean 

2000 Mathematics Subject Classification

62F12 

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Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.Universiteit AntwerpenAntwerpenBelgium
  2. 2.KU Leuven and Universiteit HasseltHasseltBelgium

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