Robustness and asymptotics of the projection median

Abstract

The projection median as introduced by Durocher and Kirkpatrick (2005); Durocher and Kirkpatrick (2009) is a robust multivariate, nonparametric location estimator. It is a weighted average of points in a sample, where each point’s weight is proportional to the fraction of directions in which that point is a univariate median. The projection median has the highest possible asymptotic breakdown and is easily approximated in any dimension. Previous works have established various geometric properties of the projection median. In this paper we examine further robustness and asymptotic properties of the projection median. We derive the influence function of the projection median which leads to bounds on the maximum bias and contamination sensitivity, as well as an exact expression for the gross error sensitivity. We discuss the degree to which the projection median satisfies these properties relative to other popular robust estimators: specifically, the Zuo projection median and the half-space median. A method for computing the robustness quantities for any distribution and dimension is provided. We then show that the projection median is strongly consistent and asymptotically normal. A method for estimating and computing the asymptotic covariance of the projection median is provided. Lastly, we introduce a large sample multivariate test of location, demonstrating the use of the aforementioned properties. We conclude that the projection median performs very well in terms of the aforementioned robustness quantities but this comes at the cost of dependence on the coordinate system as the projection median is not affine equivariant.