Discussiones Mathematicae Probability and Statistics 20(1) (2000) 25-50

AN ADAPTIVE METHOD OF ESTIMATION AND OUTLIER DETECTION IN REGRESSION APPLICABLE FOR SMALL TO MODERATE SAMPLE SIZES

Brenton R. Clarke

Mathematics and Statistics, Division of Science and Engineering
Murdoch University
Murdoch, W.A., 6150, Australia

Abstract

In small to moderate sample sizes it is important to make use of all the data when there are no outliers, for reasons of efficiency. It is equally important to guard against the possibility that there may be single or multiple outliers which can have disastrous effects on normal theory least squares estimation and inference. The purpose of this paper is to describe and illustrate the use of an adaptive regression estimation algorithm which can be used to highlight outliers, either single or multiple of varying number. The outliers can include 'bad' leverage points. Illustration is given of how 'good' leverage points are retained and 'bad' leverage points discarded. The adaptive regression estimator generalizes its high breakdown point adaptive location estimator counterpart and thus is expected to have high efficiency at the normal model. Simulations confirm this. On the other hand, examples demonstrate that the regression algorithm given highlights outliers and 'potential' outliers for closer scrutiny.

The algorithm is computer intensive for the reason that it is a global algorithm which is designed to highlight outliers automatically. This also obviates the problem of searching out ``local minima" encountered by some algorithms designed as fast search methods. Instead the objective here is to assess all observations and subsets of observations with the intention of culling all outliers which can range up to as much as approximately half the data. It is assumed that the distributional form of the data less outliers is approximately normal. If this distributional assumption fails, plots can be used to indicate such failure, and, transformations may be ;required before potential outliers are deemed as outliers. A well known set of data illustrates this point.

Keywords: outlier; least median of squares regression; least trimmed squares; trimmed likelihood; adaptive estimation; leverage.

1991 Mathematics Subject Classification: 62T05.

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Received 15 November 1998