Asymptotic formulas for the hyperġeometric function₂ F ₁ of matrix argument, useful in multivariate analysis

Summary

LetS _i have the Wishart distributionW _p(∑_i,n_i) fori=1,2. An asymptotic expansion of the distribution of$- 2\log \lambda = - 2\log \left[ {\prod\limits_{\alpha = 1}^2 {\left| {{{S_\alpha } \mathord{\left/ {\vphantom {{S_\alpha } {n_\alpha }}} \right. \kern-\nulldelimiterspace} {n_\alpha }}} \right|^{{{n_\alpha } \mathord{\left/ {\vphantom {{n_\alpha } 2}} \right. \kern-\nulldelimiterspace} 2}} } \cdot \left| {{{\left( {S_1 + S_2 } \right)} \mathord{\left/ {\vphantom {{\left( {S_1 + S_2 } \right)} n}} \right. \kern-\nulldelimiterspace} n}} \right|^{ - {n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} } \right]$ for large n=n₁+n₂ is derived, when∑ ₁∑ ⁻¹₂ =I+n^−1/2θ, based on an asymptotic solution of the system of partial differential equations for the hypergeometric function₂ F ₁, obtained recently by Muirhead [2]. Another asymptotic formula is also applied to the distributions of −2 log λ and −log|S ₂(S ₁+S ₂)⁻¹| under fixed∑ ₁∑ ⁻¹₂ , which gives the earlier results by Nagao [4]. Some useful asymptotic formulas for₁ F ₁ were investigated by Sugiura [7].