A Family of Scale Estimators by Means of Trimming

Abstract

A very common practice in literature is that of building scale estimators by means of a location estimator. The most common scale estimator used is the standard deviation, which is obtained by using the mean; its use is not the most suitable one due to its weakness under the presence of outliers. We can find other scale estimators, based on location estimators, which are more resistant under the presence of outliers, as for instance, the so called Mad, which uses for its construction the median. Since the mean and the median can be considered extreme elements of a location estimators family known as trimmed means, in this paper we propose a scale estimator family called αβ_Trimmed. For its definition, we will use the above mentioned trimmed means family and different parameters α and β, which are called trimming levels. We will demonstrate the good robustness behavior of the elements of such a family. It will be proved that they are affine equivariant, and (depending on the trimming levels) have a high exact fit point, a high breakdown point and a bounded sensitivity curve.