Testing the equality of matrix distributions

  • Lingzhe Guo
  • Reza ModarresEmail author
Original Paper


While matrices are usually used as the basic data structure for experiments with repeated measurements or longitudinal data, testing methods for the equality of two matrix distributions have not been fully discussed in the literature. In this article, we propose three methods to test the equality of two matrix distributions: the likelihood ratio test, the Frobenius norm methods and triangle tests. We present a simulation to compare their performance under the matrix normal distribution. We apply the testing methods to compare the US economy, as measured by closing prices of five market indices, before and after the US stock market crash of 2008.


Matrix distribution Matrix normal Homogeneity Frobenius norm 

Mathematics Subject Classification

62H10 62H15 62G20 



  1. Anderlucci L, Viroli C (2015) Covariance pattern mixture models for the analysis of multivariate heterogeneous longitudinal data. Ann Appl Stat 9(2):777–800MathSciNetCrossRefzbMATHGoogle Scholar
  2. Anderson TW (2003) An introduction to multivariate statistical analysis. Wiley, New YorkzbMATHGoogle Scholar
  3. Banerjee T, Firouzi H, Hero AO (2015) Non-parametric quickest change detection for large scale random matrices. In: IEEE international symposium on information theory (ISIT), 146–150Google Scholar
  4. Baringhaus L, Franz C (2004) On a new multivariate two-sample test. J Multivar Anal 88(1):190–206MathSciNetCrossRefzbMATHGoogle Scholar
  5. Biswas M, Ghosh AK (2014) A nonparametric two-sample test applicable to high dimensional data. J Multivar Anal 123:160–171MathSciNetCrossRefzbMATHGoogle Scholar
  6. Carroll JD, Arabie P (1980) Multidimensional scaling. Annu Rev Psychol 31:607649CrossRefGoogle Scholar
  7. Chen JT, Gupta AK (2005) Matrix variate skew normal distributions. Statistics 39(3):247–253MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chen J, Gupta AK (2012) Parametric statistical change point analysis: with applications to genetics, medicine, and finance. Springer, BerlinCrossRefzbMATHGoogle Scholar
  9. Dawid AP (1981) Some matrix-variate distribution theory: notational considerations and a Bayesian application. Biometrika 68:265–274MathSciNetCrossRefzbMATHGoogle Scholar
  10. Dutilleul P (1999) The MLE algorithm for the matrix normal distribution. J Stat Comput Simul 64(2):105–123CrossRefzbMATHGoogle Scholar
  11. Gallaugher MPB, McNicholas PD (2018) Finite mixtures of skewed matrix variate distributions. Pattern Recognit 80:83–93CrossRefGoogle Scholar
  12. Gallaugher MP, McNicholas PD (2017) A matrix variate skew-t distribution. Stat 6(1):160–170MathSciNetCrossRefGoogle Scholar
  13. Gupta AK, Nagar DK (1999) Matrix variate distributions, vol 104. CRC Press, FloridazbMATHGoogle Scholar
  14. Harrar SW, Gupta AK (2008) On matrix variate skew-normal distributions. Statistics 42(2):179–194MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hoeffding W, Robbins H (1948) The central limit theorem for dependent random variables. Duke Math J 15(3):773–780MathSciNetCrossRefzbMATHGoogle Scholar
  16. Liu Z, Modarres R (2011) A triangle test for equality of distribution functions in high dimensions. J Nonparametr Stat 23(3):605–615MathSciNetCrossRefzbMATHGoogle Scholar
  17. Lovison G (2006) A matrix-valued Bernoulli distribution. J Multivar Anal 97(7):1573–1585MathSciNetCrossRefzbMATHGoogle Scholar
  18. Lu N, Zimmerman DL (2005) The likelihood ratio test for a separable covariance matrix. Stat Probab Lett 73(4):449–457MathSciNetCrossRefzbMATHGoogle Scholar
  19. Maa JF, Pearl DK, Bartoszyński R (1996) Reducing multidimensional two-sample data to one-dimensional interpoint comparisons. Ann Stat 24:1069–1074MathSciNetCrossRefzbMATHGoogle Scholar
  20. Mitchell MW, Genton MG, Gumpertz ML (2006) A likelihood ratio test for separability of covariances. J Multivar Anal 97(5):1025–1043MathSciNetCrossRefzbMATHGoogle Scholar
  21. Naik DN, Rao SS (2001) Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix. J Appl Stat 28(1):91–105MathSciNetCrossRefzbMATHGoogle Scholar
  22. Roy A (2007) A note on testing of Kronecker product covariance structures for doubly multivariate data. In: Proceedings of the American Statistical Association, statistical computing section, pp 2157–2162Google Scholar
  23. Székely GJ, Rizzo ML (2004) Testing for equal distributions in high dimension. InterStat 5Google Scholar
  24. Vermunt JK (2007) A hierarchical mixture model for clustering three-way data sets. Comput Stat Data Anal 51:5368–5376MathSciNetCrossRefzbMATHGoogle Scholar
  25. Viroli C (2011) Finite mixtures of matrix normal distributions for classifying three-way data. Stat Comput 21–4:511–522MathSciNetCrossRefzbMATHGoogle Scholar
  26. Viroli C (2012) On matrix-variate regression analysis. J Multivar Anal 111:296–309MathSciNetCrossRefzbMATHGoogle Scholar
  27. Xia Y, Li L (2017) Hypothesis testing of matrix graph model with application to brain connectivity analysis. Biometrics 73(3):780–791MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of StatisticsThe George Washington UniversityWashingtonUSA

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