## The Annals of Statistics

### Laplace approximations for hypergeometric functions with matrix argument

#### Abstract

In this paper we present Laplace approximations for two functions of matrix argument: the Type I confluent hypergeometric function and the Gauss hypergeometric function. Both of these functions play an important role in distribution theory in multivariate analysis, but from a practical point of view they have proved challenging, and they have acquired a reputation for being difficult to approximate. Appealing features of the approximations we present are: (i) they are fully explicit (and simple to evaluate in practice); and (ii) typically, they have excellent numerical accuracy. The excellent numerical accuracy is demonstrated in the calculation of noncentral moments of Wilks' $\Lambda$ and the likelihood ratio statistic for testing block independence, and in the calculation of the CDF of the noncentral distribution of Wilks' $\Lambda$ via a sequential saddlepoint approximation. Relative error properties of these approximations are also studied, and it is noted that the approximations have uniformly bounded relative errors in important cases.

#### Article information

Source
Ann. Statist., Volume 30, Number 4 (2002), 1155-1177.

Dates
First available in Project Euclid: 10 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1031689021

Digital Object Identifier
doi:10.1214/aos/1031689021

Mathematical Reviews number (MathSciNet)
MR1926172

Zentralblatt MATH identifier
1029.62047

Subjects
Primary: 62H10: Distribution of statistics
Secondary: 62E17: Approximations to distributions (nonasymptotic)

#### Citation

Butler, Roland W.; Wood, Andrew T. A. Laplace approximations for hypergeometric functions with matrix argument. Ann. Statist. 30 (2002), no. 4, 1155--1177. doi:10.1214/aos/1031689021. https://projecteuclid.org/euclid.aos/1031689021

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• FORT COLLINS, COLORADO 80523 E-MAIL: walrus@stat.colostate.edu SCHOOL OF MATHEMATICAL SCIENCES UNIVERSITY OF NOTTINGHAM NOTTINGHAM NG7 2RD UNITED KINGDOM E-MAIL: atw@maths.nott.ac.uk