Adaptive Choice of Trimming

The trimmed mean is a very well known robust estimator in the location model. An outline of its history can be found in Stigler (1973). It is computationally appealing, has a simple structure, and it is easy to use for practitioners. The structure of the trimmed mean extends in a straightforward way to the trimmed LS estimator. However, one major drawback of the trimmed mean (and of the trimmed LS estimator) is that the trimming proportion α has to be fixed in advance. The proper choice of α is a natural quest ion whenever one attempts to apply these estimators. This question was studied by many statisticians who tried to determine the t rimming proportion adaptively based on the observations. Tukey and McLaughlin (1963) and later Jaeckel (1971) chose as trimming proportion the value α that minimizes the estimator of the asymptotic variance of the trimmed mean. The asymptotic behavior of this procedure was later studied by Hall (1981). In this situation, we could speak about a fully adaptive trimmed mean, which was the primary goal of these authors. Hájek (1970) proposed a simple decision procedure, based on ranks, which selects one in a finite family of distribution shapes. This could be used for t he choice of one in a finite set of α’s. Both types of adaptive procedures — the fully adaptive procedure of Tuckey-McLaughlin-Jaeckel-hall and partially adaptive procedure of Hájek — along with their extensions by Dodge and Jurečková (1997) to the linear regression model based on regression rank scores, are described in the present chapter. Some other procedures are mentionned in the notes.