Electronic Journal of Statistics

Affine-invariant rank tests for multivariate independence in independent component models

Hannu Oja, Davy Paindaveine, and Sara Taskinen

Full-text: Open access


We consider the problem of testing for multivariate independence in independent component (IC) models. Under a symmetry assumption, we develop parametric and nonparametric (signed-rank) tests. Unlike in independent component analysis (ICA), we allow for the singular cases involving more than one Gaussian independent component. The proposed rank tests are based on componentwise signed ranks, à la Puri and Sen. Unlike the Puri and Sen tests, however, our tests (i) are affine-invariant and (ii) are, for adequately chosen scores, locally and asymptotically optimal (in the Le Cam sense) at prespecified densities. Asymptotic local powers and asymptotic relative efficiencies with respect to Wilks’ LRT are derived. Finite-sample properties are investigated through a Monte-Carlo study.

Article information

Electron. J. Statist., Volume 10, Number 2 (2016), 2372-2419.

Received: January 2016
First available in Project Euclid: 6 September 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62H15: Hypothesis testing
Secondary: 62G35: Robustness

Distribution-free tests independent component models rank tests singular information matrices tests for multivariate independence uniform local asymptotic normality


Oja, Hannu; Paindaveine, Davy; Taskinen, Sara. Affine-invariant rank tests for multivariate independence in independent component models. Electron. J. Statist. 10 (2016), no. 2, 2372--2419. doi:10.1214/16-EJS1174. https://projecteuclid.org/euclid.ejs/1473187647

Export citation


  • Bauer, D. F. (1972). Constructing confidence sets using rank statistics., J. Amer. Statist. Assoc. 67 687–690.
  • Blomqvist, N. (1950). On a measure of dependence between two random variables., Ann. Math. Statist. 21 593–600.
  • Le Cam, L. (1986)., Asymptotic Methods in Statistical Decision Theory. Springer-Verlag, New York.
  • Chen, A. and Bickel, P. J. (2006). Efficient independent component analysis., Ann. Statist. 34 2825–2855.
  • Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric tests., Ann. Math. Statist. 29 972–994.
  • Dümbgen, L. (1998). On Tyler’s M-functional of scatter in high dimension., Ann. Inst. Statist. Math. 50 471–491.
  • Garel, B. and Hallin, M. (1995). Local asymptotic normality of multivariate ARMA processes with a linear trend., Ann. Inst. Statist. Math. 47 551–579.
  • Gieser, P. W. and Randles, R. H. (1997). A nonparametric test of independence between two vectors., J. Amer. Statist. Assoc. 92 561–567.
  • Hallin, M. (1994). On the Pitman-nonadmissibility of correlogram-based methods., J. Time Series Anal. 15 607–612.
  • Hallin, M. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape I: Optimal rank-based tests for sphericity., Ann. Statist. 34 2707–2756.
  • Hettmansperger, T. P. and Randles, R. H. (2002). A practical affine equivariant multivariate median., Biometrika 89 851–860.
  • Hodges, J. L. and Lehmann, E. L. (1956). The efficiency of some nonparametric competitors of the $t$-test., Ann. Math. Statist. 27 324–335.
  • Ilmonen, P. and Paindaveine, D. (2011). Semiparametrically efficient inference based on signed ranks in symmetric independent component models., Ann. Statist. 39 2448–2476.
  • Kendall, M. G. (1938). A new measure of rank correlation., Biometrika 30 81–93.
  • Lind, B. and Roussas, G. (1972). A remark on quadratic mean differentiability., Ann. Math. Statist. 43 1030–1034.
  • Möttönen, J., Hüsler, J. and Oja, H. (2003). Multivariate nonparametric tests in randomized complete block design., J. Multivariate Anal. 85 106–129.
  • Nordhausen, K., Oja, H. and Paindaveine, D. (2009). Signed-rank tests for location in the symmetric independent component model., J. Multivariate Analysis 100 821–834.
  • Nyblom, J. and Mäkeläinen, T. (1983). Comparisons of tests for the presence of random walk coefficients in a simple linear model., J. Amer. Statist. Assoc. 78 856–864.
  • Peters, D. and Randles, R. H. (1990). A multivariate signed-rank test for the one-sample location problem., J. Amer. Statist. Assoc. 85 552–557.
  • Pillai, K. C. S. (1955). Some new test criteria in multivariate analysis., Ann. Math. Statist. 26 117–121.
  • Puri, M. L. and Sen, P. K. (1971)., Nonparametric Methods in Multivariate Analysis. J. Wiley, New York.
  • Puri, M. L. and Sen, P. K. (1985)., Nonparametric Methods in General Linear Models. J. Wiley, New York.
  • Randles, R. H. (1989). A distribution-free multivariate test based on interdirections., J. Amer. Statist. Assoc. 84 1045–1050.
  • Rao, C. R. and Mitra, S. K. (1971)., Generalized Inverses of Matrices and its Applications. J. Wiley, New York.
  • Spearman, C. (1904). The proof and measurement of association between two things., Amer. J. Psychol. 15 72–101.
  • Swensen, A. R. (1985). The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend., J. Multivariate Anal. 16 54–70.
  • Taskinen, S., Kankainen, A. and Oja, H. (2003). Sign test of independence between two random vectors., Statist. Probab. Lett. 62 9–21.
  • Taskinen, S., Kankainen, A. and Oja, H. (2004). Rank scores tests of multivariate independence. In, Theory and Applications of recent Robust Methods A. S. M. Hubert, G. Pison and S. Van Aelst, eds. 329–342. Birkhauser, Basel.
  • Taskinen, S., Oja, H. and Randles, R. (2005). Multivariate nonparametric tests of independence., J. Amer. Statist. Assoc. 100 916–925.
  • Theis, F. J. (2004). A new concept for separability problems in blind source separation., Neural Comput. 16 1827–1850.
  • Tyler, D. E. (1987). A distribution-free $M$-estimator of multivariate scatter., Ann. Statist. 15 234–251.
  • Tyler, D. E., Critchley, F., Dümbgen, L. and Oja, H. (2009). Invariant co-ordinate selection., J. Roy. Statist. Soc. Ser. B 71 549–592.
  • Wilks, S. S. (1935). On the independence of $k$ sets of normally distributed statistical variables., Econometrica 3 309–326.