Strong limit theorems for empirical halfspace depth trimmed regions

2023/08/22 by Ilienko, Andrii, Molchanov, Ilya, Turin, Riccardo · 1 citation
#60F15 52A21 60D05 62H12 #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2308.11393

Abstract

We study empirical variants of the halfspace (Tukey) depth of a probability measure μ, which are obtained by replacing μ with the corresponding weighted empirical measure. We prove analogues of the Marcinkiewicz--Zygmund strong law of large numbers and of the law of the iterated logarithm in terms of set inclusions and for the Hausdorff distance between the theoretical and empirical variants of depth trimmed regions. In the special case of μ being the uniform distribution on a convex body K, the depth trimmed regions are convex floating bodies of K, and we obtain strong limit theorems for their empirical estimators.

Cited by

• Depth based trimmed means

Related

• Convergence of depths and depth-trimmed regions
• Half-space depth of log-concave probability measures
• Distributionally robust halfspace depth
• Cube root weak convergence of empirical estimators of a density level set
• Concentration of the empirical level sets of Tukey's halfspace depth