Uniform convergence rates for halfspace depth

Data depth functions are a generalization of one-dimensional order statistics and medians to real spaces of dimension greater than one; in particular, a data depth function quantifies the centrality of a point with respect to a data set or a probability distribution. One of the most commonly studied data depth functions is halfspace depth. Halfspace depth is of interest to computational geometers because it is highly geometric, and it is of interest to statisticians because it shares many desirable theoretical properties with the one-dimensional median. It is known that as the sample size increases, the halfspace depth for a sample converges to the halfspace depth for the underlying distribution, almost surely. In this paper, we use the geometry and structure of halfspace depth to reduce a high-dimensional problem into many one-dimensional problems. This bound requires only mild assumptions on the distribution, and it leads to an improved convergence rate when the underlying distribution decays exponentially, i.e., the probability that a sample point has magnitude at least R is O(exp(−λR2/2)). We also provide examples and show that our bounds are tight.