## Hiroshima Mathematical Journal

- Hiroshima Math. J.
- Volume 37, Number 1 (2007), 13-23.

### 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.

#### Article information

**Source**

Hiroshima Math. J., Volume 37, Number 1 (2007), 13-23.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.hmj/1176324092

**Digital Object Identifier**

doi:10.32917/hmj/1176324092

**Mathematical Reviews number (MathSciNet)**

MR2308521

**Zentralblatt MATH identifier**

1116.62026

**Subjects**

Primary: 62H10: Distribution of statistics

Secondary: 62E20: Asymptotic distribution theory

**Keywords**

Applications asymptotic expansions error bounds ${}_1F_1$ hypergeometric functions matrix argument

#### Citation

Fujikoshi, Yasunori. Computable error bounds for asymptotic expansions of the hypergeometric function ${}_1F_1$ of matrix argument and their applications. Hiroshima Math. J. 37 (2007), no. 1, 13--23. doi:10.32917/hmj/1176324092. https://projecteuclid.org/euclid.hmj/1176324092