## Bernoulli

• Bernoulli
• Volume 22, Number 3 (2016), 1745-1769.

### On high-dimensional sign tests

#### Abstract

Sign tests are among the most successful procedures in multivariate nonparametric statistics. In this paper, we consider several testing problems in multivariate analysis, directional statistics and multivariate time series analysis, and we show that, under appropriate symmetry assumptions, the fixed-$p$ multivariate sign tests remain valid in the high-dimensional case. Remarkably, our asymptotic results are universal, in the sense that, unlike in most previous works in high-dimensional statistics, $p$ may go to infinity in an arbitrary way as $n$ does. We conduct simulations that (i) confirm our asymptotic results, (ii) reveal that, even for relatively large $p$, chi-square critical values are to be favoured over the (asymptotically equivalent) Gaussian ones and (iii) show that, for testing i.i.d.-ness against serial dependence in the high-dimensional case, Portmanteau sign tests outperform their competitors in terms of validity-robustness.

#### Article information

Source
Bernoulli, Volume 22, Number 3 (2016), 1745-1769.

Dates
Revised: October 2014
First available in Project Euclid: 16 March 2016

https://projecteuclid.org/euclid.bj/1458132998

Digital Object Identifier
doi:10.3150/15-BEJ710

Mathematical Reviews number (MathSciNet)
MR3474832

Zentralblatt MATH identifier
1360.62225

#### Citation

Paindaveine, Davy; Verdebout, Thomas. On high-dimensional sign tests. Bernoulli 22 (2016), no. 3, 1745--1769. doi:10.3150/15-BEJ710. https://projecteuclid.org/euclid.bj/1458132998

#### References

• [1] Banerjee, A., Dhillon, I., Ghosh, J. and Sra, S. (2003). Generative model-based clustering of directional data. In Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining 19–28. New York: ACM.
• [2] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley Series in Probability and Mathematical Statistics. New York: Wiley.
• [3] Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. New York: Springer.
• [4] Cai, T., Fan, J. and Jiang, T. (2013). Distributions of angles in random packing on spheres. J. Mach. Learn. Res. 14 1837–1864.
• [5] Cai, T.T. and Jiang, T. (2012). Phase transition in limiting distributions of coherence of high-dimensional random matrices. J. Multivariate Anal. 107 24–39.
• [6] Chen, S.X., Zhang, L.-X. and Zhong, P.-S. (2010). Tests for high-dimensional covariance matrices. J. Amer. Statist. Assoc. 105 810–819.
• [7] Chikuse, Y. (1991). High-dimensional limit theorems and matrix decompositions on the Stiefel manifold. J. Multivariate Anal. 36 145–162.
• [8] Dryden, I.L. (2005). Statistical analysis on high-dimensional spheres and shape spaces. Ann. Statist. 33 1643–1665.
• [9] Dufour, J.-M., Hallin, M. and Mizera, I. (1998). Generalized runs tests for heteroscedastic time series. J. Nonparametr. Stat. 9 39–86.
• [10] Dümbgen, L. (1998). On Tyler’s $M$-functional of scatter in high dimension. Ann. Inst. Statist. Math. 50 471–491.
• [11] Frahm, G. (2004). Generalized elliptical distributions: Theory and applications. Ph.D. thesis, Universität zu Köln.
• [12] Frahm, G. and Jaekel, U. (2010). Tyler’s $M$-estimator, random matrix theory, and generalized elliptical distributions with applications to finance. Discussion paper, Department of Economic and Social Statistics, University of Cologne, Germany.
• [13] Frahm, G. and Jaekel, U. (2010). A generalization of Tyler’s $M$-estimators to the case of incomplete data. Comput. Statist. Data Anal. 54 374–393.
• [14] Hall, P., Marron, J.S. and Neeman, A. (2005). Geometric representation of high dimension, low sample size data. J. R. Stat. Soc. Ser. B. Stat. Methodol. 67 427–444.
• [15] Hallin, M. and Paindaveine, D. (2002). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30 1103–1133.
• [16] 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.
• [17] Hallin, M., Paindaveine, D. and Verdebout, T. (2010). Optimal rank-based testing for principal components. Ann. Statist. 38 3245–3299.
• [18] Jiang, T. and Yang, F. (2013). Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions. Ann. Statist. 41 2029–2074.
• [19] Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions. Vol. 2, 2nd ed. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. New York: Wiley.
• [20] Ledoit, O. and Wolf, M. (2002). Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. Ann. Statist. 30 1081–1102.
• [21] Ley, C., Swan, Y., Thiam, B. and Verdebout, T. (2013). Optimal $R$-estimation of a spherical location. Statist. Sinica 23 305–332.
• [22] Li, W.K. and McLeod, A.I. (1981). Distribution of the residual autocorrelations in multivariate ARMA time series models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 43 231–239.
• [23] Ljung, G.M. and Box, G.E. (1978). On a measure of lack of fit in time series models. Biometrika 65 297–303.
• [24] Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Berlin: Springer.
• [25] Mardia, K.V. and Jupp, P.E. (2000). Directional Statistics. Wiley Series in Probability and Statistics. Chichester: Wiley.
• [26] Möttönen, J. and Oja, H. (1995). Multivariate spatial sign and rank methods. J. Nonparametr. Stat. 5 201–213.
• [27] Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. New York: Wiley.
• [28] Oja, H. (2010). Multivariate Nonparametric Methods with R: An Approach Based on Spatial Signs and Ranks. Lecture Notes in Statistics 199. New York: Springer.
• [29] Onatski, A., Moreira, M.J. and Hallin, M. (2013). Asymptotic power of sphericity tests for high-dimensional data. Ann. Statist. 41 1204–1231.
• [30] Paindaveine, D. (2009). On multivariate runs tests for randomness. J. Amer. Statist. Assoc. 104 1525–1538.
• [31] Paindaveine, D. and Verdebout, T. (2014). Optimal rank-based tests for the location parameter of a rotationally symmetric distribution on the hypersphere. In Mathematical Statistics and Limit Theorems: Festschrift in Honor of Paul Deheuvels (M. Hallin, D. Mason, D. Pfeifer and J. Steinebach, eds.) 249–270. Berlin: Springer.
• [32] Paindaveine, D. and Verdebout, T. (2015). Supplement to “On high-dimensional sign tests.” DOI:10.3150/15-BEJ710SUPP.
• [33] Randles, R.H. (1989). A distribution-free multivariate sign test based on interdirections. J. Amer. Statist. Assoc. 84 1045–1050.
• [34] Randles, R.H. (2000). A simpler, affine-invariant, multivariate, distribution-free sign test. J. Amer. Statist. Assoc. 95 1263–1268.
• [35] Rayleigh, L. (1919). On the problem of random vibrations and random flights in one, two and three dimensions. Phil. Mag. 37 321–346.
• [36] Sirkiä, S., Taskinen, S., Oja, H. and Tyler, D.E. (2009). Tests and estimates of shape based on spatial signs and ranks. J. Nonparametr. Stat. 21 155–176.
• [37] Srivastava, M.S. and Fujikoshi, Y. (2006). Multivariate analysis of variance with fewer observations than the dimension. J. Multivariate Anal. 97 1927–1940.
• [38] Srivastava, M.S. and Kubokawa, T. (2013). Tests for multivariate analysis of variance in high dimension under non-normality. J. Multivariate Anal. 115 204–216.
• [39] Taskinen, S., Kankainen, A. and Oja, H. (2003). Sign test of independence between two random vectors. Statist. Probab. Lett. 62 9–21.
• [40] Taskinen, S., Oja, H. and Randles, R.H. (2005). Multivariate nonparametric tests of independence. J. Amer. Statist. Assoc. 100 916–925.
• [41] Tyler, D.E. (1987). A distribution-free $M$-estimator of multivariate scatter. Ann. Statist. 15 234–251.
• [42] Zou, C., Peng, L., Feng, L. and Wang, Z. (2014). Multivariate sign-based high-dimensional tests for sphericity. Biometrika 101 229–236.

#### Supplemental materials

• Supplement to “High-dimensional sign tests”. The supplement article contains the proofs of Theorems 2.4 and 2.5 together with simulation results related to the sign test for independence. It also provides histograms from the simulations of Section 3.1.