Electronic Journal of Statistics

On the mean and variance of the generalized inverse of a singular Wishart matrix

R. Dennis Cook and Liliana Forzani

Full-text: Open access

Abstract

We derive the first and the second moments of the Moore-Penrose generalized inverse of a singular standard Wishart matrix without relying on a density. Instead, we use the moments of an inverse Wishart distribution and an invariance argument which is related to the literature on tensor functions. We also find the order of the spectral norm of the generalized inverse of a Wishart matrix as its dimension and degrees of freedom diverge.

Article information

Source
Electron. J. Statist., Volume 5 (2011), 146-158.

Dates
First available in Project Euclid: 15 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1300198786

Digital Object Identifier
doi:10.1214/11-EJS602

Mathematical Reviews number (MathSciNet)
MR2786485

Zentralblatt MATH identifier
1274.62350

Subjects
Primary: 62H05: Characterization and structure theory
Secondary: 62E15: Exact distribution theory

Keywords
Inverse Wishart distribution Moore-Penrose generalized inverse singular inverse Wishart distributions tensor functions

Citation

Cook, R. Dennis; Forzani, Liliana. On the mean and variance of the generalized inverse of a singular Wishart matrix. Electron. J. Statist. 5 (2011), 146--158. doi:10.1214/11-EJS602. https://projecteuclid.org/euclid.ejs/1300198786


Export citation

References

  • [1] Basser, P. J. and Pajevic, S. (2007). Spectral decomposition of a 4th order covariance tensor: Applications to diffusion tensor MRI., Signal Processing 87, 220–236.
  • [2] Bodnar, T. and Okhrin, Y. (2008). Properties of the singular, inverse and generalized inverse partitioned Wishart distributions., J. Multivariate Anal. 99, 2389–2405.
  • [3] Dauxois, J., Romain, Y. and Viguier-Pla, S. (1994). Tensor products and statistics., Linear Algebra and its Applications 210, 59–88.
  • [4] Díaz-García, J. A. and Gutiérrez-Jáimez, R. (2006). Distribution of the generalized inverse of a random matrix and its applications., Journal of Statistical Planning and Inference 136, 183–192.
  • [5] Eaton, M. L. (2007)., Multivariate Statistics: A Vector Space Approach. Beachwood, Ohio: Institute of Mathematical Statistics.
  • [6] Jeffreys, H. (1931)., Cartesian Tensors. Cambridge: Cambridge University Press. Page 91.
  • [7] Jog, C. S. (2006). A concise proof of the representation theorem for fourth-order isotropic tensors., J. Elasticity 85, 119–124.
  • [8] Henderson, H, V. and Searle, S. R. (1979). Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics., Canadian J. Statist. 7, 65–81.
  • [9] Itskov, M. (2009)., Tensor Analysis and Tensor Algebra for Engineers, 2nd Edition. New York: Springer.
  • [10] Magnus, J. R. and Neudecker, H. (1979). The commutation matrix: Some properties and applications., Ann. Statist. 7, 381–394.
  • [11] Neudecker, H. and Wansbeek, T. (1983). Some Results on Commutation Matrices, with Statistical Applications., Canadian J. Statist. 11, 221–231.
  • [12] Ogden, R. W. (2001). Elements of the theory of finite elasticity. In Fu, Y. B. and Ogden, R. W. (eds), Nonlinear Elasticity: Theory and Applications. Cambridge University Press, 1–58.
  • [13] Stone, M. (1987)., Coordinate-Free Multivariate Statistics. New York: Oxford University Press.
  • [14] Srivastava, M. S. (2003). Singular Wishart and multivariate beta distributions., Ann. Statist. 31, 1537–1560.
  • [15] Tyler, D. E. (1981). Asymptotic inference for eigenvectors., Ann. Statist. 9, 725–736.
  • [16] Uhlig, H. (1994). On singular Wishart and singular multivariate beta distributions., Ann. Statist. 22, 395–405.
  • [17] von Rosen, D. (1988). Moments for the inverted Wishart distribution., Scand. J. Statist. 15, 97–109.
  • [18] Zhang, Z. (2007). Pseudo-inverse multivariate/matrix-variate distributions., J. Multivariate Anal. 98, 1684–1692.