Finite Sample Breakdown Points of Projection Based Multivariate Location and Scatter Statistics

Abstract

Finite sample breakdown points are obtained for two classes of projection based multivariate location and scatter statistics: the Stahel-Donoho statistics and the Maronna-Yohai statistics. The definition of these multivariate statistics are dependent on the value of location and scale statistics for all univariate projections of the data, and consequently their properties depend on the nature of the corresponding univariate location and scale statistics used on the projected data. The finite sample breakdown points of the multivariate statistics, though, are not directly related to those of the corresponding univariate location and scale statistics. A uniform finite sample breakdown point concept is needed. The median and the median absolute deviation about the median (M.A.D.) are one possible choice for the univariate location and scale statistics, respectively. For sparse data sets in high dimensions, though, it is not recommended that the M.A.D. be used as the univariate scale statistic for the projected data since its uniform finite sample breakdown point is shown to be much less than optimum. A simple modification to the M.A.D., however, is shown to alleviate this problem. For various reasons, one may wish to consider univariate location and scale statistics other than the median and the M.A.D., respectively. A very broad and natural class of univariate location and scale statistics to consider for the projected data is the simultaneous $M$-estimates of location and scale. New results on their breakdown properties are given in this paper. Implicit formulas for the breakdown points of monotonic simultaneous $M$-estimates of location and scale are known, and they tend to imply rather low breakdown points for smooth choices of the defining weight functions. It is shown here that this phenomenon does not occur for a large class of nonmonotonic simultaneous $M$-estimates of location and scale. Furthermore, explicit rather than implicit expressions for the uniform finite sample breakdown points are given for these nonmonotonic $M$-estimates.