Computable error bounds for asymptotic expansions of the hypergeometric function ${}_1F_1$ of matrix argument and their applications

Abstract

In this paper we derive error bounds for asymptotic expansions of the hypergeometric functions ${}_1F_1(n; n+b; Z)$ and ${}_1F_1(n; n+b; -Z)$, where $Z$ is a $p \times p$ symmetric nonnegative definite matrix. The first result is applied for theoretical accuracy of approximating the moments of $\Lambda=|S_e|/|S_e+S_h|$, where $S_h$ and $S_e$ are independently distributed as a noncentral Wishart distribution $W_p(q, \Sigma, \Sigma^{1/2} \Omega \Sigma^{1/2})$ and a central Wishart distribution $W_p(n, \Sigma)$, respectively. The second result is applied for theoretical accuracy of approximating the probability density function of the maximum likelihood estimators of regression coefficients in the growth curve model.