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August 2002 Laplace approximations for hypergeometric functions with matrix argument
Roland W. Butler, Andrew T. A. Wood
Ann. Statist. 30(4): 1155-1177 (August 2002). DOI: 10.1214/aos/1031689021


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.


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Roland W. Butler. Andrew T. A. Wood. "Laplace approximations for hypergeometric functions with matrix argument." Ann. Statist. 30 (4) 1155 - 1177, August 2002.


Published: August 2002
First available in Project Euclid: 10 September 2002

zbMATH: 1029.62047
MathSciNet: MR1926172
Digital Object Identifier: 10.1214/aos/1031689021

Primary: 62H10
Secondary: 62E17

Keywords: confluent hypergeometric function , Gauss hypergeometric function , Laplace approximation , likelihood ratio test , matrix-argument hypergeometric function , saddlepoint approximation , sequential saddlepoint approximation , Wilks' $\Lambda$

Rights: Copyright © 2002 Institute of Mathematical Statistics

Vol.30 • No. 4 • August 2002
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