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Zbl 1134.33315
Richards, Donald St.P.
Total positivity properties of generalized hypergeometric functions of matrix argument. (English)
[J] J. Stat. Phys. 116, No. 1-4, 907-922 (2004). ISSN 0022-4715; ISSN 1572-9613/e

Summary: In multivariate statistical analysis, several authors have studied the total positivity properties of the generalized $(_{0} F _{1})$ 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 $_{0} F _{1}$ functions 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 $TP_{2}$properties; 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).

MSC 2000:: *33C70 Other hypergeometric functions and integrals in several variables
33C20 Generalized hypergeometric series
33E20 Functions defined by series and integrals
43A90 Spherical functions (abstract harmonic analysis)
62H99 Multivariate analysis
60E15 Inequalities in probability theory
15A52 Random matrices

Keywords: 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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