zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Laplace approximation for Bessel functions of matrix argument. (English) Zbl 1027.65033
The authors present Laplace approximations to Bessel functions of matrix argument and to the confluent type II matrix argument hypergeometric function. A collection of results concerning the theoretical properties of the various Laplace approximations is proved. In particular, it is shown that suitably calibrated versions of the approximations inherit the symmetries and confluence relations satisfied by the original functions. A series of tables with results of calculations is given.

65D20Computation of special functions, construction of tables
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
33C15Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
33F05Numerical approximation and evaluation of special functions
Full Text: DOI
[1] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions. (1972) · Zbl 0543.33001
[2] Anderson, G. A.: An asymptotic expansion for the distribution of the latent roots of the estimate covariance matrix. Ann. math. Statist. 36, 1153-1173 (1970) · Zbl 0134.15001
[3] Bleistein, N.; Handelsman, R. A.: Asymptotic expansions of integrals. (1975) · Zbl 0327.41027
[4] Butler, R. W.: Extendibility and the optimality of F, T2, and forward variable selection. Scand. J. Statist. 13, 257-262 (1986) · Zbl 0627.62064
[5] Butler, R. W.: Generalized inverse Gaussian distributions and their Wishart connections. Scand. J. Statist. 25, 69-75 (1998) · Zbl 0909.60022
[6] R.W. Butler, A.T.A. Wood, Laplace approximations to hypergeometric functions of two matrix arguments, 2001, submitted for publication. · Zbl 1075.62015
[7] Butler, R. W.; Wood, A. T. A.: Laplace approximations for hypergeometric functions with matrix argument. Ann. statist. 30, 1155-1177 (2002) · Zbl 1029.62047
[8] M.L. Eaton, On some multivariate distributions and random orthogonal matrices, in: L.M. LeCam, R.A. Olshen (Eds.), Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. II, Wadsworth, Inc., Belman, CA, 1985.
[9] Fisher, R. A.: Statistical methods, experimental design, and scientific inference. (1990) · Zbl 0705.62003
[10] Goodall, C. R.; Mardia, K. V.: Multivariate aspects of shape theory. Ann. statist. 21, 848-866 (1993) · Zbl 0788.62045
[11] Herz, C. S.: Bessel functions of matrix argument. Ann. math. 61, 474-523 (1955) · Zbl 0066.32002
[12] Khatri, C. G.; Mardia, K. V.: The von-Mises--Fisher matrix distribution in orientation statistics. J. roy. Statist. soc. Ser. B 39, 95-106 (1977) · Zbl 0356.62044
[13] Magnus, J. R.: Linear structures. (1988) · Zbl 0667.15010
[14] Magnus, J. R.; Neudecker, H.: Matrix differential calculus with appl. Statist. econometric. (1988) · Zbl 0651.15001
[15] Mathai, A. M.: A handbook of generalized special functions for statistical and physical sciences. (1993) · Zbl 0770.33001
[16] Muirhead, R. J.: Latent roots and matrix variatesa review of some asymptotic results. Ann. statist. 6, 5-33 (1978) · Zbl 0375.62050
[17] Muirhead, R. J.: Aspects of multivariate statistical theory. (1982) · Zbl 0556.62028
[18] Prentice, M. J.: Orientation statistics without parametric assumptions. J. roy. Statist. soc. Ser. B 48, 214-222 (1986) · Zbl 0601.62071
[19] Tierney, L.; Kadane, J. B.: Accurate approximations for posterior moments and marginal densities. J. amer. Statist. assoc. 81, 82-86 (1986) · Zbl 0587.62067
[20] Wood, A. T. A.: Estimation of the concentration parameters of the Fisher matrix distribution on $SO(3)$ and the Bingham distribution on sq,q $\geqslant $2. Austral. J. Statist. 35, 69-79 (1993) · Zbl 0772.62029