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Zbl 1029.62047
Butler, Ronald W.; Wood, Andrew T.A.
Laplace approximations for hypergeometric functions with matrix argument. (English)
[J] Ann. Stat. 30, No. 4, 1155-1177 (2002). ISSN 0090-5364

Summary: We present Laplace approximations for two functions of matrix arguments: 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 non-central moments of Wilks' $\Lambda$ and the likelihood ratio statistic for testing block independence, and in the calculation of the CDF of the non-central distribution of Wilks' $\Lambda$ via a sequential saddle-point 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.

MSC 2000:: *62H10 Multivariate distributions of statistics
33C99 Hypergeometric functions
33C15 Confluent hypergeometric functions
62E17 Approximations to statistical distributions (nonasymptotic)

Keywords: Confluent hypergeometric function; Gauss hypergeometric function; Laplace approximation; likelihood ratio test; matrix-argument

Cited in: Zbl 1075.62015

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