M-estimation: Some Remedies

Abstract

This paper is concerned with two problems facing M- estimators. M-estimators bound the influence of large residuals, or more generally, large deviations from the mean. In doing so, however, they can become inconsistent, in particular in the case of non-Normal Generalized Linear Models (GLMs), thus leading to biased estimates. We present a method for correcting such biased estimates without needing to alter the standard estimation procedures used in most statistical packages. Another problem facing M-estimators is that they do not take into account the (potentially high) leverage of a point. The cause of high leverage could be the mis-recording of a single explanatory variate value. We investigate the Mean Shift Outlier Model (MSOM) as discussed by Cook and Weisberg (1982), and show that this can lead to a method for reducing the leverage of a suspect point. However, we also note that in other cases, the method would increase the leverage (albeit removing the influence) and therefore some other approach is needed. Our discussion is based on the Normal case, although we have in mind extensions of these proposals in the GLM context.