Laplace approximations for hypergeometric functions with matrix argument

• 成果类型: Article
• 署名作者: Butler, RW; Wood, ATA
• 署名单位: Colorado State University System; Colorado State University Fort Collins; University of Nottingham
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2002
• 页码: 1155-1177
• 关键词: saddlepoint approximations; distributions

摘要

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' A and the likelihood ratio statistic for testing block independence, and in the calculation of the CDF of the noncentral distribution of Wilks' A 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.