Semiparametrically Efficient Tests of Multivariate Independence Using Center-Outward Quadrant, Spearman, and Kendall Statistics

Defining multivariate generalizations of the classical univariate ranks has been a long-standing open problem in statistics. Optimal transport has been shown to offer a solution in which multivariate ranks are obtained by transporting data points to a grid that approximates a uniform reference measure (Chernozhukov et al., 2017; Hallin, 2017; Hallin et al., 2021). We take up this new perspective to develop and study multivariate analogues of the sign covariance/quadrant statistic, Kendall’s tau, and Spearman’s rho. The resulting tests of multivariate independence are genuinely distribution-free, hence uniformly valid irrespective of the actual (absolutely continuous) distributions of the observations. Our results provide asymptotic distribution theory for these new test statistics, with asymptotic approximations to critical values to be used for testing independence as well as a power analysis of the resulting tests. This includes a multivariate elliptical Chernoff–Savage property, which guarantees that, under ellipticity, our nonparametric tests of independence enjoy an asymptotic relative efficiency of one or larger with respect to the classical Gaussian procedures.

[1]  Han Liu,et al.  Rank-based testing for semiparametric VAR models: A measure transportation approach , 2020, Bernoulli.

[2]  E. Barrio,et al.  Nonparametric Multiple-Output Center-Outward Quantile Regression , 2022, 2204.11756.

[3]  Daniel Hlubinka,et al.  Efficient Fully Distribution-Free Center-Outward Rank Tests for Multiple-Output Regression and MANOVA , 2020, Journal of the American Statistical Association.

[4]  M. Drton,et al.  On universally consistent and fully distribution-free rank tests of vector independence , 2020, The Annals of Statistics.

[5]  B. Sen,et al.  Multivariate ranks and quantiles using optimal transport: Consistency, rates and nonparametric testing , 2019, The Annals of Statistics.

[6]  Marc Hallin,et al.  On the Finite-Sample Performance of Measure Transportation-Based Multivariate Rank Tests , 2021, 2111.04705.

[7]  M. Hallin.  Measure Transportation and Statistical Decision Theory , 2021, Annual Review of Statistics and Its Application.

[8]  B. Sen,et al.  Efficiency Lower Bounds for Distribution-Free Hotelling-Type Two-Sample Tests Based on Optimal Transport , 2021, 2104.01986.

[9]  Carlos Matrán,et al.  Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach , 2021, The Annals of Statistics.

[10]  Haeun Moon,et al.  Interpoint-ranking sign covariance for the test of independence , 2021, Biometrika.

[11]  Bodhisattva Sen,et al.  Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation , 2019, Journal of the American Statistical Association.

[12]  Marc Hallin,et al.  A note on the regularity of optimal-transport-based center-outward distribution and quantile functions , 2020, J. Multivar. Anal..

[13]  Han Liu,et al.  Center-Outward R-Estimation for Semiparametric VARMA Models , 2019, Journal of the American Statistical Association.

[14]  M. Drton,et al.  Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs , 2019, Journal of the American Statistical Association.

[15]  E. Barrio,et al.  Distribution and Quantile Functions, Ranks, and Signs in Rd: a measure transportation approach , 2020 .

[16]  J. A. Cuesta-Albertos,et al.  Smooth Cyclically Monotone Interpolation and Empirical Center-Outward Distribution Functions , 2018 .

[17]  A. Figalli.  On the continuity of center-outward distribution and quantile functions , 2018, Nonlinear Analysis.

[18]  N. Meinshausen,et al.  Symmetric rank covariances: a generalized framework for nonparametric measures of dependence , 2017, Biometrika.

[19]  Guillaume Carlier,et al.  Vector quantile regression beyond the specified case , 2017, J. Multivar. Anal..

[20]  V. Chernozhukov,et al.  Monge-Kantorovich Depth, Quantiles, Ranks and Signs , 2014, 1412.8434.

[21]  G. Carlier,et al.  Vector Quantile Regression , 2014, 1406.4643.

[22]  C. Spearman.  The proof and measurement of association between two things. , 2015, International journal of epidemiology.

[23]  H. Oja.  Multivariate Nonparametric Methods with R: An approach based on spatial signs and ranks , 2010 .

[24]  D. Paindaveine,et al.  Chernoff-Savage and Hodges-Lehmann Results for Wilks' Test of Multivariate Independence , 2008, 0805.2305.

[25]  Xuming He,et al.  On the limiting distributions of multivariate depth-based rank sum statistics and related tests , 2006, 0708.0167.

[26]  R. Randles,et al.  Multivariate Nonparametric Tests of Independence , 2005 .

[27]  D. Paindaveine.  A unified and elementary proof of serial and nonserial, univariate and multivariate: Chernoff-Savage results , 2004 .

[28]  Emanuel Parzen,et al.  Quantile Probability and Statistical Data Modeling , 2004 .

[29]  H. Oja,et al.  Rank Scores Tests of Multivariate Independence , 2004 .

[30]  H. Oja,et al.  Sign test of independence between two random vectors , 2003 .

[31]  D. Paindaveine,et al.  Optimal procedures based on interdirections and pseudo-Mahalanobis ranks for testing multivariate elliptic white noise against ARMA dependence , 2002 .

[32]  D. Paindaveine,et al.  Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks , 2002 .

[33]  R. Randles,et al.  A Nonparametric Test of Independence between Two Vectors , 1997 .

[34]  R. McCann.  Existence and uniqueness of monotone measure-preserving maps , 1995 .

[35]  M. Hallin,et al.  Local asymptotic normality of multivariate ARMA processes with a linear trend , 1995, Annals of the Institute of Statistical Mathematics.

[36]  Regina Y. Liu,et al.  A Quality Index Based on Data Depth and Multivariate Rank Tests , 1993 .

[37]  R. Randles.  A Distribution-Free Multivariate Sign Test Based on Interdirections , 1989 .

[38]  A. Barbour,et al.  Random association of symmetric arrays , 1986 .

[39]  Bruce A. Lind,et al.  A Remark on Quadratic Mean Differentiability , 1972 .

[40]  P. Sen,et al.  Nonparametric methods in multivariate analysis , 1974 .

[41]  P. Sen,et al.  Theory of rank tests , 1969 .

[42]  Herman Chernoff,et al.  ASYMPTOTIC NORMALITY AND EFFICIENCY OF CERTAIN NONPARAMETRIC TEST STATISTICS , 1958 .

[43]  E. J. Hannan,et al.  The Asymptotic Powers of Certain Tests Based on Multiple Correlations , 1956 .

[44]  J. L. Hodges,et al.  The Efficiency of Some Nonparametric Competitors of the t-Test , 1956 .

[45]  H. S. Konijn.  On the Power of Certain Tests for Independence in Bivariate Populations , 1956 .

[46]  W. Hoeffding.  A Combinatorial Central Limit Theorem , 1951 .

[47]  Nils Blomqvist,et al.  On a Measure of Dependence Between two Random Variables , 1950 .

[48]  W. Hoeffding.  A Class of Statistics with Asymptotically Normal Distribution , 1948 .

[49]  F. Wilcoxon.  Individual Comparisons by Ranking Methods , 1945 .

[50]  M. Kendall.  A NEW MEASURE OF RANK CORRELATION , 1938 .

[51]  S. S. Wilks.  On the Independence of k Sets of Normally Distributed Statistical Variables , 1935 .