The Multivariate $S_n$ Estimator

Abstract

In this note we introduce the M$S_n$ estimator (for Multivariate $S_n$) a new robust estimator of multivariate ranking. Like MVE and MCD it searches for an $h$-subset which minimizes a criterion. The difference is that the new criterion measures the degree of overlap between univariate projections of the data. A primary advantage of this new criterion lies in its relative independence from the configuration of the outliers. A second advantage is that it easily lends itself to so-called "symmetricizing" transformations whereby the observations only enter the objective function through their pairwise differences: this makes our proposal well suited for models with an asymmetric distribution. M$S_n$ is, therefore, more generally applicable than either MVE, MCD or SDE. We also construct a fast algorithm for the M$S_n$ estimator, and simulate its bias under various adversary configurations of outliers.