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