Multivariate goodness-of-fit tests based on Wasserstein distance

Abstract

Goodness-of-fit tests based on the empirical Wasserstein distance are proposed for simple and composite null hypotheses involving general multivariate distributions. For group families, the procedure is to be implemented after preliminary reduction of the data via invariance. This property allows for calculation of exact critical values and p-values at finite sample sizes. Applications include testing for location-scale families and testing for families arising from affine transformations, such as elliptical distributions with given standard radial density and unspecified location vector and scatter matrix. A novel test for multivariate normality with unspecified mean vector and covariance matrix arises as a special case. For more general parametric families, we propose a parametric bootstrap procedure to calculate critical values. The lack of asymptotic distribution theory for the empirical Wasserstein distance means that the validity of the parametric bootstrap under the null hypothesis remains a conjecture. Nevertheless, we show that the test is consistent against fixed alternatives. To this end, we prove a uniform law of large numbers for the empirical distribution in Wasserstein distance, where the uniformity is over any class of underlying distributions satisfying a uniform integrability condition but no additional moment assumptions. The calculation of test statistics boils down to solving the well-studied semi-discrete optimal transport problem. Extensive numerical experiments demonstrate the practical feasibility and the excellent performance of the proposed tests for the Wasserstein distance of order p = 1 and p = 2 and for dimensions at least up to d = 5. The simulations also lend support to the conjecture of the asymptotic validity of the parametric bootstrap.

Empirical Optimal Transport between Different Measures Adapts to Lower Complexity

“…With recent computational advances (for a survey see Bertsimas & Tsitsiklis 1997;Peyré & Cuturi 2019) OT based methodology is also quickly emerging as a useful tool for data analysis with diverse applications in statistics. This includes bootstrap and resampling (Bickel & Freedman, 1981;Sommerfeld et al, 2019;Heinemann et al, 2020), goodness of fit testing (del Barrio et al, 1999;Hallin et al, 2021b), multivariate quantiles and ranks (Chernozhukov et al, 2017;Hallin et al, 2021a) and general notions of dependency (Nies et al, 2021;Mordant & Segers, 2022). For a recent survey see Panaretos & Zemel (2019).…”

WATCH: Wasserstein Change Point Detection for High-Dimensional Time Series Data

“…WATCH is implemented in Python v3.9 and numpy v1.19.5. We also used the R implementation of the Wasserstein distance provided in [40], and called it from the Python code using the rpy2 v3.4.5 bridge. All experiments are run on a machine with an Intel Core i7-8750H CPU, GeForce GTX 1050 Ti Mobile GPU and 32 GB of RAM.…”

“…According to the Wasserstein distance definition in [40], let P (R d ) be the set of Borel probability measures on R d and P p (R d ) be the subset of such measures with a finite moment of order p ∈ [1, inf). For P, Q ∈ P (R d ), let Γ(P, Q) be the set of probability measures γ on R d × R d with marginals P and Q, i.e., such that γ(B × R d ) = P(B) and In terms of random variables X and Y with laws P and Q, respectively, the p-Wasserstein distance is the smallest value of {E(dist(X − Y )) p } 1 P over all possible joint distributions γ ∈ Γ(P, Q) of (X, Y).…”

“…Intuitively, given a k-dimensional vector space V ⊂ R k , we can map the distributions (P, Q) to corresponding sets of k-dimensional data points G, H ⊆ V . In our work, the Wasserstein distance W P is calculated using the SAG approximated computational method as explained in [40]. From now on, we refer to the approximated version of the Wasserstein distance as W A ≈ W p .…”

Measure Transportation and Statistical Decision Theory

“…Based on the techniques developed for testing vector independence, Ghosal and Sen (2019) and Deb and Sen (2019) also propose rank-based tests for the two-sample goodness-of-fit problem (see also Boeckel et al (2018) and Hallin et al (2021b)) and the null hypothesis of symmetry. The spirit of their approach is quite similar to that developed for testing vector independence, with (for Deb and Sen (2019)) tests based on energy statistics (Székely and Rizzo, 2013) rather than distance covariance; we refer to the papers for further details.…”