Trimmed slope estimates for simple linear regression

Abstract

For the simple linear regression model, the trimmed slope estimate introduced here is a direct extension of the concept of a trimmed mean. To calculate a trimmed mean, order the observations, disregard unusually high and low ones, and take an average of the remaining observations. Specifying which observations are unusual is the discriminating factor among several types of trimmed means. Similar to the mean, the least squares estimate of the slope may be thought of as a weighted average of slopes calculated from pairs of observations. To calculate a trimmed slope estimate, order the pairwise slopes, disregard unusually high and low ones, and take a weighted average of the remaining slopes. In this paper we discuss various methods of specifying the unusual slopes to be trimmed. We also allow a broad class of weights to be used in the averaging. In some situations which admit outliers in the data, from a robustness viewpoint certain choices of weights are useful even without trimming pairwise slopes. Trimmed slope estimates enjoy advantages similar to those of trimmed means. They are easy to interpret and explain. Since they are linear combinations of the pairwise slopes, computation of the estimates is straightforward. Further, the data analyst can control the amount of trimming through an explicit parameter. It turns out that the large sample properties are also similar. In this paper the consistency, asymptotic normality and influence functions of the trimmed slope estimators are investigated. Show more