[Paper Review] Minimax Estimation of the Volume of a Set with Smooth Boundary

Ery Arias-Castro, Beatriz Pateiro‐López+1 more|arXiv (Cornell University)|May 4, 2016

Statistical Methods and Inference20 references3 citations

TL;DR

This paper proposes a minimax-optimal estimator for the volume of a compact set with smooth boundary in $\mathbb{R}^d$, using a data-splitting approach that corrects bias in the $\alpha$-convex hull plug-in estimator. By leveraging the $r$-rolling condition (equivalent to positive reach), the method achieves the minimax rate $n^{-(d+3)/(2d+2)}$, matching a derived lower bound.

ABSTRACT

We consider the problem of estimating the volume of a compact domain in a Euclidean space based on a uniform sample from the domain. We assume the domain has a boundary with positive reach. We propose a data splitting approach to correct the bias of the plug-in estimator based on the sample alpha-convex hull. We show that this simple estimator achieves a minimax lower bound that we derive. Some numerical experiments corroborate our theoretical findings.

Motivation & Objective

• To derive a minimax lower bound for volume estimation over compact sets with smooth boundary in $\mathbb{R}^d$.
• To develop a bias-corrected volume estimator that achieves this minimax rate.
• To establish theoretical guarantees under the $r$-rolling condition, equivalent to positive reach of the boundary.
• To provide a practical estimator with confidence interval extension and numerical validation.

Proposed method

• The estimator splits the sample into two subsamples: one for constructing the $\alpha$-convex hull $\hat{S}$, and another for estimating the volume of $S \setminus \hat{S}$.
• The volume estimator is defined as $\hat{V} = \mu(\hat{S}) / \max(1/2, 1 - \hat{p})$, where $\hat{p}$ is the proportion of second-sample points outside $\hat{S}$.
• The method relies on the $r$-rolling condition, ensuring that both $S$ and its complement admit balls of radius $r$ touching the boundary.
• Theoretical analysis uses unavoidable families of balls to bound the probability of empty balls in the sample, enabling concentration bounds.
• The estimator is shown to achieve the minimax rate $n^{-(d+3)/(2d+2)}$ under the assumption $\alpha \in (0,r]$ and $\beta \leq m/n \leq 1 - \beta$.
• The approach enables construction of confidence intervals and is validated via numerical experiments.

Experimental results

Research questions

• RQ1 What is the minimax rate for estimating the volume of a compact set with smooth boundary in $\mathbb{R}^d$?
• RQ2 Can a bias-corrected estimator based on the $\alpha$-convex hull achieve this minimax rate?
• RQ3 How does the $r$-rolling condition (positive reach) affect the convergence rate of volume estimators?
• RQ4 What is the role of data splitting in reducing bias in volume estimation from uniform samples?
• RQ5 Can the proposed estimator be extended to provide valid confidence intervals?

Key findings

• The paper establishes a minimax lower bound of order $C\delta^2 n^{-(d+3)/(2d+2)}$ for volume estimation over sets satisfying the $r$-rolling condition.
• The proposed estimator $\hat{V}$ achieves the minimax rate $n^{-(d+3)/(2d+2)}$, matching the lower bound.
• The estimator is robust under the assumption $\alpha \in (0,r]$, ensuring the $\alpha$-convex hull captures the true set structure.
• Theoretical analysis shows that the error in the $\alpha$-convex hull estimator decays as $O(\textnormal{e}^{-L_1 n \alpha_n^d} + \alpha_n^{-(d-1)/(d+1)} n^{-2/(d+1)})$, with the second term dominating.
• The data-splitting strategy effectively reduces bias in the plug-in estimator, which otherwise only achieves $O(n^{-2/(d+1)})$ rate.
• Numerical experiments corroborate the theoretical findings, showing improved convergence over the standard plug-in estimator.