Resampling simplicial depth

2026-08-11

AI Summary

This study addresses the statistical inference challenge posed by the sample simplicial depth (SD) under an unknown center of symmetry, where the convergence rate—either √n or n—is uncertain, leading to ambiguity between degenerate and non-degenerate U-statistic behavior. To resolve this, the authors propose a two-stage adaptive subsampling procedure: first estimating the convergence rate parameter γ ∈ {1/2, 1} via bias-corrected subsampling, then using this estimate to approximate the sampling distribution of SD. This approach provides the first unified framework that adaptively handles both degenerate and non-degenerate cases in a data-driven manner, with rigorous theoretical guarantees of consistency. Both theoretical analysis and empirical experiments demonstrate that the method effectively constructs valid confidence intervals for SD and substantially enhances its practical utility and stability in supervised classification tasks.

Abstract

The simplicial depth (SD) is a commonly used indicator of the centrality of points $x\in\mathbb{R}^d$ with respect to distributions $P$ on $\mathbb{R}^d$. Asymptotic theory for the sample SD is based on its representation as a $U$-statistic, which can be either non-degenerate or degenerate. For $d=2$, we prove under mild conditions that this $U$-statistic is degenerate with rate $n$ if and only if $x$ is a center of symmetry of $P$. Otherwise, the asymptotic distribution of SD is non-degenerate with rate $\sqrt{n}$. Because the location of the center of symmetry of $P$ is usually unknown, these two modes of behavior complicate the estimation of the sample distribution of SD at $x$. We propose a two-step adaptive subsampling procedure for estimating that distribution. First, an estimator $\widehat γ$ of a parameter $γ\in\{1/2, 1\}$ characterizing the correct rate of convergence $n^γ$ of SD is constructed based on subsampling. Our estimator uses a bias correction suitable for $U$-statistics. Second, $\widehat γ$ is employed for approximating the distribution of the sample SD. We prove the consistency of this subsampling approach and illustrate its usefulness (i) in the construction of confidence intervals for SD, and (ii) in an SD-based supervised classification task

Problem

• simplicial depth
• U-statistic
• asymptotic distribution
• subsampling
• rate of convergence

Innovation

• simplicial depth
• U-statistic
• subsampling
• adaptive estimation
• degeneracy