Laplace approximation for Bessel functions of matrix argument

Authors:
Ronald W. Butler;Andrew T. A. Wood
Affiliations:
Department of Statistics, Colorado State University, Fort Collins, CO;School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Venue:
Journal of Computational and Applied Mathematics
Year:
2003

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Abstract

We derive Laplace approximations to three functions of matrix argument which arise in statistics and elsewhere: matrix Bessel Av; matrix Bessel Bv; and the type II confluent hypergeometric function of matrix argument, Ψ. We examine the theoretical and numerical properties of the approximations. On the theoretical side, it is shown that the Laplace approximations to Av, Bv and Ψ given here, together with the Laplace approximations to the matrix argument functions 1F1 and 2F1 presented in Butler and Wood (Laplace approximations to hyper-geometric functions with matrix argument, Ann. Statist. (2002)), satisfy all the important confluence relations and symmetry relations enjoyed by the original functions.