Two-Sample Inference for Gaussian-Smoothed Wasserstein Costs with Finite Moments

TL;DR

This paper analyzes the convergence rates and statistical properties of a two-sample estimator for Gaussian-smoothed Wasserstein costs under finite moments, providing bounds, CLTs, and variance estimators.

Contribution

It establishes new probabilistic bounds, asymptotic expansions, and inference tools for Gaussian-smoothed Wasserstein costs with finite moments.

Findings

Upper bounds in probability for the estimator's error rate.

A first-order expansion and CLT for the Wasserstein cost when p>1.

A sample-splitting variance estimator for the two-sample setting.

Abstract

We study the two-sample plug-in estimator of the Gaussian-smoothed Wasserstein cost \(T_p^{(\sigma)}(\mu,\nu)=W_p(\mu*\gamma_\sigma,\nu*\gamma_\sigma)^p\) on \(\R^d\). For fixed smoothing and finite polynomial moments \(M_{q_\mu}(\mu)<\infty\), \(M_{q_\nu}(\nu)<\infty\), with \(q_\mu,q_\nu>p\), we establish upper bounds in probability of order \(\rho_{q_\mu,p,d}(m)+\rho_{q_\nu,p,d}(n)\). Here \(\rho_{q,p,d}(N)=N^{-(q-p)/(q+d)}\) for \(p<q<d+2p\), \(N^{-1/2}\log N\) at \(q=d+2p\), and \(N^{-1/2}\) for \(q>d+2p\). This order also holds in expectation under \(q_\mu,q_\nu\ge2p\). When the smoothed population distance is positive, the cost bound yields this rate for the distance itself. For \(p>1\) and \(q_\mu,q_\nu>d+2p\), we also derive a first-order expansion, a separated two-sample central limit theorem, and a sample-splitting variance estimator.