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Laplace approximations for hypergeometric functions with matrix argument. (English) Zbl 1029.62047

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’ Λ and the likelihood ratio statistic for testing block independence, and in the calculation of the CDF of the non-central distribution of Wilks’ Λ 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.

62H10Multivariate distributions of statistics
33C99Hypergeometric functions
33C15Confluent hypergeometric functions, Whittaker functions, 1 F 1
62E17Approximations to statistical distributions (nonasymptotic)