The Breakdown Points of the Mean Combined With Some Rejection Rules

TLDR

Outlier rejection methods have historically been studied without considering their quantitative impact on estimation or testing, and no comparison with other robust techniques had been made until recently; general properties and relationships to other robust methods are also discussed. The article investigates estimation of a location parameter in the presence of outliers using a Monte Carlo study. The study compares the mean after outlier rejection with other robust location estimators, provides formulas for the breakdown points of six rejection rules, and illustrates the approach with real‑data examples. Monte Carlo variances of the arithmetic mean under various rejection rules were computed, and the results are succinctly explained by the breakdown points of the combined procedures, which also clarify and replace the masking effect concept.

Abstract

In the past, methods for rejection of outliers have been investigated mostly without regard to the quantitative consequences for subsequent estimation or testing procedures. Moreover, although rejection of outliers with subsequent application of least squares methods is one of the oldest and most widespread classes of robust procedures, until recently no comparison was made with other robust methods. In this article the simplest situation, namely estimation of a location parameter in the potential presence of outliers, is treated by means of a Monte Carlo study. This study yields Monte Carlo variances of the "arithmetic mean" after rejection of outliers according to several classical and recent formal rules. The results are also compared with those for other robust estimators of location parameters. It turns out that a simple summary and theoretical explanation of the Monte Carlo results is provided by the breakdown points of the combined rejection-estimation procedures. As a by-product, the concept of breakdown point also leads to a better understanding of the so-called "masking effect" and can in fact replace the latter concept. Formulas for the breakdown points are given for the six types of rejection rules used. Some general aspects and properties of all methods for the rejection of outliers and their relation to other robust methods are also discussed. Finally, the treatment of outliers in the context of real data is considered, and several examples are briefly mentioned; one real-life example is analyzed in greater detail.