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.
"Laplace approximations for hypergeometric functions with matrix argument." Ann. Statist. 30 (4) 1155 - 1177, August 2002. https://doi.org/10.1214/aos/1031689021