Integral transform methods in goodness-of-fit testing, II: the Wishart distributions

  • Elena Hadjicosta
  • Donald RichardsEmail author


We initiate the study of goodness-of-fit testing for data consisting of positive definite matrices. Motivated by the appearance of positive definite matrices in numerous applications, including factor analysis, diffusion tensor imaging, volatility models for financial time series, wireless communication systems, and polarimetric radar imaging, we apply the method of Hankel transforms of matrix argument to develop goodness-of-fit tests for Wishart distributions with given shape parameter and unknown scale matrix. We obtain the limiting null distribution of the test statistic and a corresponding covariance operator, show that the eigenvalues of the operator satisfy an interlacing property, and apply our test to some financial data. We establish the consistency of the test against a large class of alternative distributions and derive the asymptotic distribution of the test statistic under a sequence of contiguous alternatives. We obtain the Bahadur and Pitman efficiency properties of the test statistic and establish a modified version of Wieand’s condition.


Bahadur slope Bessel function of matrix argument Contiguous alternative Diffusion tensor imaging Factor analysis Gaussian random field Pitman efficiency Zonal polynomial 



We thank the reviewers and the editors for their constructive comments.

Supplementary material

10463_2019_737_MOESM1_ESM.pdf (702 kb)
Supplementary material 1 (pdf 701 KB)


  1. Anfinsen, S. N., Doulgeris, A. P., Eltoft, T. (2011). Goodness-of-fit tests for multilook polarimetric radar data based on the Mellin transform. IEEE Transactions on Geoscience and Remote Sensing, 49, 2764–2781.CrossRefGoogle Scholar
  2. Anfinsen, S. N., Eltoft, T. (2011). Application of the matrix-variate Mellin transform to analysis of polarimetric radar images. IEEE Transactions on Geoscience and Remote Sensing, 49, 2281–2295.CrossRefGoogle Scholar
  3. Asai, M., McAleer, M., Yu, J. (2006). Multivariate stochastic volatility: A review. Econometric Reviews, 25, 145–175.CrossRefMathSciNetzbMATHGoogle Scholar
  4. Bahadur, R. R. (1960). Stochastic comparison of tests. Annals of Mathematical Statistics, 31, 276–295.CrossRefMathSciNetzbMATHGoogle Scholar
  5. Bahadur, R. R. (1967). Rates of convergence of estimates and test statistics. Annals of Mathematical Statistics, 38, 303–324.CrossRefMathSciNetzbMATHGoogle Scholar
  6. Bahadur, R. R. (1971). Some limit theorems in statistics. Philadelphia, PA: SIAM.CrossRefzbMATHGoogle Scholar
  7. Baringhaus, L., Ebner, B., Henze, N. (2017). The limit distribution of weighted \(L^2\)-goodness-of-fit statistics under fixed alternatives, with applications. Annals of the Institute of Statistical Mathematics, 69, 969–995.CrossRefMathSciNetzbMATHGoogle Scholar
  8. Baringhaus, L., Taherizadeh, F. (2010). Empirical Hankel transforms and their applications to goodness-of-fit tests. Journal of Multivariate Analysis, 101, 1445–1467.CrossRefMathSciNetzbMATHGoogle Scholar
  9. Baringhaus, L., Taherizadeh, F. (2013). A K-S type test for exponentiality based on empirical Hankel transforms. Communications in Statistics – Theory and Methods, 42, 3781–3792.CrossRefMathSciNetzbMATHGoogle Scholar
  10. Bauer, H. (1981). Probability theory and elements of measure theory, second English edition. New York, NY: Academic Press.Google Scholar
  11. Billingsley, P. (1968). Convergence of probability measures. New York: Wiley.zbMATHGoogle Scholar
  12. Billingsley, P. (1979). Probability and measure. New York: Wiley.zbMATHGoogle Scholar
  13. Bishop, A. N., Del Moral, P., Niclas, A. (2017). An introduction to Wishart matrix moments. Foundations and Trends in Machine Learning, 11, 97–218.CrossRefzbMATHGoogle Scholar
  14. Brislawn, C. (1991). Traceable integral kernels on countably generated measure spaces. Pacific Journal of Mathematics, 150, 229–240.CrossRefMathSciNetzbMATHGoogle Scholar
  15. Browne, M. W. (1968). A comparison of factor analytic techniques. Psychometrika, 33, 267–334.CrossRefMathSciNetGoogle Scholar
  16. Burkill, J. C., Burkill, H. (2002). A second course in mathematical analysis. New York: Cambridge University Press.zbMATHGoogle Scholar
  17. Butler, R. W. (1998). Generalized inverse Gaussian distributions and their Wishart connections. Scandinavian Journal of Statistics, 25, 69–75.CrossRefMathSciNetzbMATHGoogle Scholar
  18. Constantine, A. G. (1966). The distribution of Hotelling’s generalised \(T_0^2\). Annals of Mathematical Statistics, 37, 215–225.CrossRefMathSciNetGoogle Scholar
  19. Damon, B. M., Ding, Z., Anderson, A. W., Freyer, A. S., Gore, J. C. (2002). Validation of diffusion tensor MRI-based muscle fiber tracking. Magnetic Resonance in Medicine, 48, 97–104.CrossRefGoogle Scholar
  20. Dancis, J., Davis, C. (1987). An interlacing theorem for eigenvalues of self-adjoint operators. Linear Algebra and its Applications, 88(89), 117–122.CrossRefMathSciNetzbMATHGoogle Scholar
  21. Del Moral, P., Niclas, A. (2018). A Taylor expansion of the square root matrix functional. Journal of Mathematical Analysis and Applications, 465, 259–266.CrossRefMathSciNetzbMATHGoogle Scholar
  22. Dryden, I. L., Koloydenko, A., Zhou, D. (2009). Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Annals of Applied Statistics, 3, 1102–1123.CrossRefMathSciNetzbMATHGoogle Scholar
  23. Faraut, J., Korányi, A. (1994). Analysis on symmetric cones. Oxford: Oxford University Press.zbMATHGoogle Scholar
  24. Farrell, R. H. (1985). Multivariate calculation. New York: Springer.CrossRefzbMATHGoogle Scholar
  25. G\({\bar{\i }}\)khman, Ĭ. \(\bar{\rm I}\)., Skorokhod, A. V. (1980). Thetheory of stochastic processes, Vol. 1. New York: Springer.Google Scholar
  26. Goodman, N. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction). The Annals of Mathematical Statistics, 34, 152–177.CrossRefMathSciNetzbMATHGoogle Scholar
  27. Gourieroux, C., Sufana, R. (2010). Derivative pricing with Wishart multivariate stochastic volatility. Journal of Business & Economic Statistics, 28, 438–451.CrossRefMathSciNetzbMATHGoogle Scholar
  28. Gross, K. I., Richards, D. St. P. (1987). Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Transactions of the American Mathematical Society, 301, 781–811.Google Scholar
  29. Gupta, R. D., Richards, D. St. P. (1987). Multivariate Liouville distributions. Journal of Multivariate Analysis, 23, 233–256.Google Scholar
  30. Gupta, R. D., Richards, D. St. P. (1995). Multivariate Liouville distributions, IV. Journal of Multivariate Analysis, 54, 1–17.Google Scholar
  31. Hadjicosta, E. (2019). Integral transform methods in goodness-of-fit testing. Doctoral dissertation, Pennsylvania State University, University Park.Google Scholar
  32. Hadjicosta, E., Richards, D. (2019). Integral transform methods in goodness-of-fit testing, I: The gamma distributions. Metrika (to appear) (Preprint, arXiv:1810.07138).
  33. Haff, L. R., Kim, P. T., Koo, J. Y., Richards, D. St. P. (2011). Minimax estimation for mixtures of Wishart distributions. Annals of Statistics, 39, 3417–3440.Google Scholar
  34. Henze, N., Meintanis, S. G., Ebner, B. (2012). Goodness-of-fit tests for the gamma distribution based on the empirical Laplace transform. Communications in Statistics — Theory & Methods, 41, 1543–1556.CrossRefMathSciNetzbMATHGoogle Scholar
  35. Herz, C. S. (1955). Bessel functions of matrix argument. Annals of Mathematics, 61, 474–523.CrossRefMathSciNetzbMATHGoogle Scholar
  36. Hochstadt, H. (1973). One-dimensional perturbations of compact operators. Proceedings of the American Mathematical Society, 37, 465–467.CrossRefMathSciNetzbMATHGoogle Scholar
  37. Horn, R. A., Johnson, C. R. (1990). Matrix analysis. New York: Cambridge University Press.zbMATHGoogle Scholar
  38. Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables. Biometrika, 48, 419–426.CrossRefMathSciNetzbMATHGoogle Scholar
  39. James, A. T. (1964). Distributions of matrix variates and latent roots derived from normal samples. The Annals of Mathematical Statistics, 35, 475–501.CrossRefMathSciNetzbMATHGoogle Scholar
  40. Jian, B., Vemuri, B. C. (2007). Multi-fiber reconstruction from diffusion MRI using mixture of Wisharts and sparse deconvolution. Information Processing in Medical Imaging, 20, 384–395.CrossRefGoogle Scholar
  41. Jian, B., Vemuri, B. C., Özarslan, E., Carney, P. R., Mareci, T. H. (2007). A novel tensor distribution model for the diffusion-weighted MR signal. NeuroImage, 37, 164–176.CrossRefGoogle Scholar
  42. Kågström, B. (1977). Bounds and perturbation bounds for the matrix exponential. BIT Numerical Mathematics, 17, 39–57.CrossRefMathSciNetzbMATHGoogle Scholar
  43. Khatri, C. G. (1966). On certain distribution problems based on positive definite quadratic functions in normal vectors. The Annals of Mathematical Statistics, 37, 468–479.CrossRefMathSciNetzbMATHGoogle Scholar
  44. Kim, P. T., Richards, D. St. P. (2011). Deconvolution density estimation on the space of positive definite symmetric matrices. In D. Hunter, et al. (Eds.), Nonparametric statistics and mixture models, pp. 58–68. Singapore: World Scientific Press.Google Scholar
  45. Koev, P., Edelman, A. (2006). The efficient evaluation of the hypergeometric function of a matrix argument. Mathematics of Computation, 75, 833–846.CrossRefMathSciNetzbMATHGoogle Scholar
  46. Kotz, S., Johnson, N. L., Boyd, D. W. (1967). Series representations of distributions of quadratic forms in normal variables. I. Central case. Annals of Mathematical Statistics, 38, 823–837.CrossRefMathSciNetzbMATHGoogle Scholar
  47. Ku, Y. C., Bloomfield, P. (2010). Generating random Wishart matrices with fractional degrees of freedom in OX. Preprint: North Carolina State University, Raleigh.Google Scholar
  48. Le Maître, O. P., Knio, O. M. (2010). Spectral methods for uncertainty quantification. New York: Springer.CrossRefzbMATHGoogle Scholar
  49. Ledoux, M., Talagrand, M. (1991). Probability in Banach spaces. New York: Springer.CrossRefzbMATHGoogle Scholar
  50. Lee, H. N., Schwartzman, A. (2017). Inference for eigenvalues and eigenvectors in exponential families of random symmetric matrices. Journal of Multivariate Analysis, 162, 152–171.CrossRefMathSciNetzbMATHGoogle Scholar
  51. Maass, H. (1971). Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol 216. New York: Springer.Google Scholar
  52. Matsui, M., Takemura, A. (2008). Goodness-of-fit tests for symmetric stable distributions–Empirical characteristic function approach. TEST, 17, 546–566.CrossRefMathSciNetzbMATHGoogle Scholar
  53. Matthews, P. M., Arnold, D. L. (2001). Magnetic resonance imaging of multiple sclerosis: New insights linking pathology to clinical evolution. Current Opinion in Neurology, 14, 279–287.CrossRefGoogle Scholar
  54. Muirhead, R. J. (1982). Aspects of multivariate statistical theory. New York: Wiley.CrossRefzbMATHGoogle Scholar
  55. Neumann-Haefelin, T., Moseley, M. E., Albers, G. W. (2000). New magnetic resonance imaging methods for cerebrovascular disease: Emerging clinical applications. Annals of Neurology, 47, 559–570.CrossRefGoogle Scholar
  56. Pomara, N., Crandall, D. T., Choi, S. J., Johnson, G., Lim, K. O. (2001). White matter abnormalities in HIV-1 infection: A diffusion tensor imaging study. Psychiatry Research: Neuroimaging, 106, 15–24.Google Scholar
  57. Richards, D. St. P. (2010). Chapter 35: Functions of Matrix Argument. In F. W. Olver, D. W. Lozier, R. F. Boisvert & C. W. Clark (Eds.), NIST handbook of mathematical functions. New York: Cambridge University Press.Google Scholar
  58. Rosenbloom, M., Sullivan, E. V., Pfefferbaum, A. (2003). Using magnetic resonance imaging and diffusion tensor imaging to assess brain damage in alcoholics. Alcohol Research and Health, 27, 146–152.Google Scholar
  59. Schwartzman, A. (2006). Random ellipsoids and false discovery Rates: Statistics for diffusion tensor imaging data, Doctoral dissertation, Stanford University, Palo Alto.Google Scholar
  60. Schwartzman, A., Dougherty, R. F., Taylor, J. E. (2005). Cross-subject comparison of principal diffusion direction maps. Magnetic Resonance in Medicine, 53, 1423–1431.CrossRefGoogle Scholar
  61. Schwartzman, A., Dougherty, R. F., Taylor, J. E. (2008). False discovery rate analysis of brain diffusion direction maps. The Annals of Applied Statistics, 2, 153–175.CrossRefMathSciNetzbMATHGoogle Scholar
  62. Severini, T. A. (2005). Elements of distribution theory. New York: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  63. Shilov, G. E. (1977). Linear algebra. New York: Dover.Google Scholar
  64. Siriteanu, C., Kuriki, S., Richards, D., Takemura, A. (2016). Chi-square mixture representations for the distribution of the scalar Schur complement in a noncentral Wishart matrix. Statistics and Probability Letters, 115, 79–87.CrossRefMathSciNetzbMATHGoogle Scholar
  65. Siriteanu, C., Takemura, A., Kuriki, S., Richards, D., Shin, H. (2015). Schur complement based analysis of MIMO zero-forcing for Rician fading. IEEE Transactions on Wireless Communications, 14, 1757–1771.CrossRefGoogle Scholar
  66. Taherizadeh, F. (2009). Empirical Hankel transform and statistical goodness-of-fit tests for exponential distributions, Doctoral dissertation, Leibniz Universität Hannover, Hannover.Google Scholar
  67. Tulino, A. M., Verdú, S. (2004). Random matrix theory and wireless communications. Hanover, MA: Now Publishers.CrossRefzbMATHGoogle Scholar
  68. Wieand, H. S. (1976). A condition under which the Pitman and Bahadur approaches to efficiency coincide. Annals of Statistics, 4, 1003–1011.CrossRefMathSciNetzbMATHGoogle Scholar
  69. Wihler, T. P. (2009). On the Hölder continuity of matrix functions for normal matrices. Journal of Inequalities in Pure and Applied Mathematics, 10, 1–5.MathSciNetzbMATHGoogle Scholar
  70. Young, N. (1998). An introduction to Hilbert space. New York: Cambridge University Press.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2019

Authors and Affiliations

  1. 1.Department of StatisticsPennsylvania State UniversityUniversity ParkUSA

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