## The Annals of Statistics

### Distance multivariance: New dependence measures for random vectors

#### Abstract

We introduce two new measures for the dependence of $n\ge2$ random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted $L^{2}$-distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Székely, Rizzo and Bakirov) from pairs of random variables to $n$-tuplets of random variables. We show that total distance multivariance can be used to detect the independence of $n$ random variables and has a simple finite-sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.

#### Article information

Source
Ann. Statist., Volume 47, Number 5 (2019), 2757-2789.

Dates
Revised: September 2018
First available in Project Euclid: 3 August 2019

https://projecteuclid.org/euclid.aos/1564797863

Digital Object Identifier
doi:10.1214/18-AOS1764

Mathematical Reviews number (MathSciNet)
MR3988772

#### Citation

Böttcher, Björn; Keller-Ressel, Martin; Schilling, René L. Distance multivariance: New dependence measures for random vectors. Ann. Statist. 47 (2019), no. 5, 2757--2789. doi:10.1214/18-AOS1764. https://projecteuclid.org/euclid.aos/1564797863

#### References

• [1] Bakirov, N. K. and Székely, G. J. (2010). Brownian covariance central limit theorem for stationary sequences. Teor. Veroyatn. Primen. 55 462–488. All references to this paper are based on the english translation (Nail K. Bakirov and Gábor J. Székely. Brownian covariance and central limit theorem for stationary sequences. Theory of Probability & Its Applications, 55(3):371–394, 2011).
• [2] Bartholomäus, J. (2018). Private communication, TU Dresden.
• [3] Berg, C. and Forst, G. (1975). Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 87. Springer, New York.
• [4] Berrett, T. B. and Samworth, R. J. (2017). Nonparametric independence testing via mutual information. Available at arXiv:1711.06642v1.
• [5] Berschneider, G. and Böttcher, B. (2018). On complex Gaussian random fields, Gaussian quadratic forms and sample distance multivariance. Available at arXiv:1808.07280v1.
• [6] Böttcher, B. (2017). Dependence structures—estimation and visualization using distance multivariance. Available at arXiv:1712.06532v1.
• [7] Böttcher, B. (2017). multivariance: Measuring multivariate dependence using distance multivariance. R package version 1.0.5.
• [8] Böttcher, B., Keller-Ressel, M. and Schilling, R. L. (2018). Detecting independence of random vectors: Generalized distance covariance and Gaussian covariance. Mod. Stoch. Theory Appl. 5 353–383.
• [9] Böttcher, B., Keller-Ressel, M. and Schilling, R. L. (2019). Supplement to “Distance multivariance: New dependence measures for random vectors.” DOI:10.1214/18-AOS1764SUPP.
• [10] Chow, Y. S. and Teicher, H. (1997). Probability Theory: Independence, Interchangeability, Martingales, 3rd ed. Springer Texts in Statistics. Springer, New York.
• [11] Comon, P. (1994). Independent component analysis, a new concept? Signal Process. 36 287–314.
• [12] Cope, L. (2009). Discussion of: Brownian distance covariance [MR2752127]. Ann. Appl. Stat. 3 1279–1281.
• [13] Csörgő, S. (1985). Testing for independence by the empirical characteristic function. J. Multivariate Anal. 16 290–299.
• [14] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II. 2nd ed. Wiley, New York.
• [15] Gretton, A., Bousquet, O., Smola, A. and Schölkopf, B. (2005). Measuring statistical dependence with Hilbert–Schmidt norms. In Algorithmic Learning Theory. Lecture Notes in Computer Science 3734 63–77. Springer, Berlin.
• [16] Jacob, N. (2001). Pseudo Differential Operators and Markov Processes. Vol. I: Fourier Analysis and Semigroups. Imperial College Press, London.
• [17] Jin, Z. and Matteson, D. S. (2018). Generalizing distance covariance to measure and test multivariate mutual dependence via complete and incomplete V-statistics. J. Multivariate Anal. 168 304–322. Available at arXiv:1709.02532. All references to this paper are based on the arxiv version 1709.02532v5 which differs from the published version.
• [18] Pfister, N., Bühlmann, P., Schölkopf, B. and Peters, J. (2018). Kernel-based tests for joint independence. J. R. Stat. Soc. Ser. B. Stat. Methodol. 80 5–31.
• [19] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge. Translated from the 1990 Japanese original, revised by the author.
• [20] Schilling, R. L. (2017). Measures, Integrals and Martingales, 2nd ed. Cambridge Univ. Press, Cambridge.
• [21] Sejdinovic, D., Sriperumbudur, B., Gretton, A. and Fukumizu, K. (2013). Equivalence of distance-based and RKHS-based statistics in hypothesis testing. Ann. Statist. 41 2263–2291.
• [22] Székely, G. J. and Bakirov, N. K. (2003). Extremal probabilities for Gaussian quadratic forms. Probab. Theory Related Fields 126 184–202.
• [23] Székely, G. J. and Rizzo, M. L. (2009). Brownian distance covariance. Ann. Appl. Stat. 3 1236–1265.
• [24] Székely, G. J. and Rizzo, M. L. (2009). Rejoinder: Brownian distance covariance. Ann. Appl. Stat. 3 1303–1308.
• [25] Székely, G. J., Rizzo, M. L. and Bakirov, N. K. (2007). Measuring and testing dependence by correlation of distances. Ann. Statist. 35 2769–2794.
• [26] Yao, S., Zhang, X. and Shao, X. (2018). Testing mutual independence in high dimension via distance covariance. J. R. Stat. Soc. Ser. B. Stat. Methodol. 80 455–480.