Journal of Statistical Physics

, Volume 116, Issue 1–4, pp 907–922 | Cite as

Total Positivity Properties of Generalized Hypergeometric Functions of Matrix Argument

  • Donald St. P. Richards


In multivariate statistical analysis, several authors have studied the total positivity properties of the generalized (0F1) hypergeometric function of two real symmetric matrix arguments. In this paper, we make use of zonal polynomial expansions to obtain a new proof of a result that these 0F1functions fail to satisfy certain pairwise total positivity properties; this proof extends both to arbitrary generalized ( r F s ) functions of two matrix arguments and to the generalized hypergeometric functions of Hermitian matrix arguments. In the case of the generalized hypergeometric functions of two Hermitian matrix arguments, we prove that these functions satisfy certain modified pairwise TP2properties; the proofs of these results are based on Sylvester's formula for compound determinants and the condensation formula of C. L. Dodgson [Lewis Carroll] (1866).

compound determinant condensation formula FKG inequality likelihood ratio test statistics monotone power function random matrix total positivity noncentral Wishart distribution zonal polynomial 


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Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • Donald St. P. Richards
    • 1
  1. 1.Department of StatisticsPenn State University

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