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).

The cited supplement is included in the PDF file

References in corpus (1)

• Multidimensional trimming based on projection depth

Cited by in corpus (8)

• Data depth and floating body
• On general notions of depth for regression
• Halfspace depth for general measures: The ray basis theorem and its consequences
• Some multivariate goodness of fit tests based on data depth
• From Halfspace M-depth to Multiple-output Expectile Regression
• Dirichlet Depths for Point Process
• A Pseudo-Metric between Probability Distributions based on Depth-Trimmed Regions
• Estimation of the covariate conditional tail expectation : a depth-based level set approach