Laplace approximations for hypergeometric functions with matrix argument

Butler, Roland W. ; Wood, Andrew T. A.

Ann. Statist., Tome 30 (2002) no. 1, p. 1155-1177 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2002-08-14
Classification:  Confluent hypergeometric function,  Gauss hypergeometric function,  Laplace approximation,  likelihood ratio test,  matrix-argument hypergeometric function,  saddlepoint approximation,  sequential saddlepoint approximation,  Wilks' $\Lambda$,  62H10,  62E17