Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation

By Nabarun Deb, Bodhisattva Sen. Journal of the American Statistical Association, 2019

175 citations recorded by the source corpus.

Abstract In this article, we propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of multivariate ranks defined using the theory of measure transportation. Unlike other existing proposals in the literature, these multivariate ranks share a number of useful properties with the usual one-dimensional ranks; most importantly, these ranks are distribution-free. This crucial observation allows us to design nonparametric tests that are exactly distribution-free under the null hypothesis. We demonstrate the applicability of this approach by constructing exact distribution-free tests for two classical nonparametric problems: (I) testing for mutual independence between random vectors, and (II) testing for the equality of multivariate distributions. In particular, we propose (multivariate) rank versions of distance covariance and energy statistic for testing scenarios (I) and (II), respectively. In both these problems, we derive the asymptotic null distribution of the proposed test statistics. We further show that our tests are consistent against all fixed alternatives. Moreover, the proposed tests are computationally feasible and are well-defined under minimal assumptions on the underlying distributions (e.g., they do not need any moment assumptions). We also demonstrate the efficacy of these procedures via extensive simulations. In the process of analyzing the theoretical properties of our procedures, we end up proving some new results in the theory of measure transportation and in the limit theory of permutation statistics using Stein’s method for exchangeable pairs, which may be of independent interest.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/bffb23af543fc65f440f2c48a49576260ad60fd4

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Citations in the corpus

A New Coefficient of Correlation (2019, background): “There have been many proposals to address this deficiency of the classical coefficients [23], such as the maximal correlation coefficient [5, 15, 21, 33], various coefficients based on joint cumulative distribution functions and ranks [3, 4, 8, 9, 11, 14, 18, 22, 30, 32, 35, 36, 46–49], kernel-based”
Stability of entropic optimal transport and Schrödinger bridges (2021, methodology): “Multivariate ranks have been defined in nonparametric statistics through Brenier’s optimal transport map to extend the usual scalar notions and tests; see [15, 19, 20, 27].”
On the power of Chatterjee’s rank correlation (2020, background): “This makes any subsequent local power analysis extremely difficult, as also observed in a parallel study of center-outward rank and sign–based tests performed in Shi et al. (2020a) and Shi et al. (2020b) (see, also, Deb and Sen (2019)).”
Measuring Association on Topological Spaces Using Kernels and Geometric Graphs (2020, methodology): “In Section 8 we use the recently proposed idea of multivariate ranks based on the theory of optimal transport (see [50, 51, 25, 30, 89]) to develop measures of association that satisfy properties (I)-(III) and are also finite-sample distribution-free when μ = μX ⊗ μY ; see Sections 8.”
On Azadkia–Chatterjee’s conditional dependence coefficient (2021, methodology): “To test the independence between Y ∈ R1 and W ∈ Rp+q, we will adopt the test proposed in Shi et al. (2022a, Equation (13)); see Deb and Sen (2021) for a similar result.”
Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs (2019, methodology): “…differences in the precise setup of multivariate ranks: while we base ourselves directly on recent work by Hallin (2017) and by Figalli (2018), Deb and Sen (2019) considered transports to the unit cube rather than the unit ball (see Definition 2.2) and present weakened assumptions in the…”
Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach (2021, background): “Deb and Sen (2019) obtain similar results using different reference uniform distributions, different empirical transports, and different asymptotic techniques.”