重抽样单纯形深度
Resampling simplicial depth
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中文总结 AI 辅助
针对二维情况下样本单纯形深度（SD）的渐近分布因对称中心未知而复杂的问题，提出带偏差校正的两步自适应子抽样法，实现分布估计并应用于置信区间构建与监督分类。
中文摘要 AI 辅助
单纯形深度（Simplicial Depth，简称SD）是衡量ℝᵈ中点x相对于ℝᵈ上分布P的中心性的常用指标。样本SD的渐近理论基于其作为U统计量的表示，该统计量可分为非退化或退化两类。对于d=2，我们在温和条件下证明：当且仅当x是P的对称中心时，该U统计量以速率n退化；否则，SD的渐近分布为非退化，速率为√n。由于P的对称中心位置通常未知，这两种行为模式会使x处SD的样本分布估计变得复杂。我们提出一种两步自适应子抽样程序来估计该分布：首先，基于子抽样构造参数γ∈{1/2,1}的估计量γ̂，该参数表征SD的正确收敛速率n^γ，我们的估计量采用适用于U统计量的偏差校正；其次，利用γ̂近似样本SD的分布。我们证明了该子抽样方法的一致性，并通过两方面示例说明其实用性：（i）构建SD的置信区间；（ii）基于SD的监督分类任务。
英文摘要
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
