Sample Out-Of-Sample Inference Based on Wasserstein Distance

TL;DR

This paper introduces SOS inference, a new method based on Wasserstein distance for out-of-sample analysis, useful in various applications like DRO, with unique asymptotic properties differing from empirical likelihood.

Contribution

The paper proposes a novel SOS inference approach using Wasserstein distance, expanding the toolkit for out-of-sample and distributionally robust analysis with new asymptotic insights.

Findings

Wasserstein-based SOS inference differs qualitatively from empirical likelihood.

Asymptotic distributions are often not chi-squared, unlike EL.

Convergence rates depend on dimension but remain manageable.

Abstract

We present a novel inference approach that we call Sample Out-of-Sample (or SOS) inference. The approach can be used widely, ranging from semi-supervised learning to stress testing, and it is fundamental in the application of data-driven Distributionally Robust Optimization (DRO). Our method enables measuring the impact of plausible out-of-sample scenarios in a given performance measure of interest, such as a financial loss. The methodology is inspired by Empirical Likelihood (EL), but we optimize the empirical Wasserstein distance (instead of the empirical likelihood) induced by observations. From a methodological standpoint, our analysis of the asymptotic behavior of the induced Wasserstein-distance profile function shows dramatic qualitative differences relative to EL. For instance, in contrast to EL, which typically yields chi-squared weak convergence limits, our asymptotic distributions are often not chi-squared. Also, the rates of convergence that we obtain have some dependence on the dimension in a non-trivial way but remain controlled as the dimension increases.