On min-max majority and deepest points

The asymptotic properties of a multivariate location estimator are obtained in this paper. The estimator examined is based on the notion of half-space depth, where the depth of a point is the minimum probability content of all half spaces containing the point. The location estimator of interest is the deepest point with respect to the empirical measure on half spaces. For angularly symmetric distributions, this estimator is consistent. For two dimensions, the exact limit distribution is derived, and the extension of the limit distribution results to higher dimensions is discussed.