Convergence of depths and depth-trimmed regions

Abstract: Depth is a concept that measures the `centrality' of a point in a given data cloud or in a given probability distribution. Every depth defines a family of so-called trimmed regions. For statistical applications it is desirable that with increasing sample size the empirical depth as well as the empirical trimmed regions converge almost surely to their population counterparts. In this article the connections between different types of convergence are discussed. In particular, conditions are given under which the pointwise (resp. uniform) convergence of the data depth implies the pointwise (resp. compact) convergence of the trimmed regions in the Hausdorff metric as well as conditions under which the reverse implications hold. Further, it is shown that under relative weak conditions the pointwise convergence of the data depth (resp. trimmed regions) is equivalent to the uniform convergence of the data depth (resp. compact convergence of the trimmed regions).