Sample Out-of-Sample Inference Based on Wasserstein Distance

We present a novel inference approach that we call sample out-of-sample inference. The approach can be used widely, ranging from semisupervised learning to stress testing, and it is fundamental in the application of data-driven distributionally robust optimization. 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 nontrivial way but remain controlled as the dimension increases.