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October, 1968 Three Multidimensional-integral Identities with Bayesian Applications
James M. Dickey
Ann. Math. Statist. 39(5): 1615-1628 (October, 1968). DOI: 10.1214/aoms/1177698143


The first identity (Section 2) expresses a moment of a product of multivariate $t$ densities as an integral of dimension one less than the number of factors. This identity is applied to inference concerning the location parameters of a multivariate normal distribution. The second identity (Section 3) expresses the density of a linear combination of independently distributed multivariate $t$ vectors, a multivariate Behrens-Fisher density (Cornish (1965)), as an integral of dimension one less than the number of summands. The two-summand version of the second identity is essentially equivalent to the two-factor version of the first identity. A synthetic representation is given for the random vector, generalizing Ruben's (1960) representation in the univariate case. The second identity is applied to multivariate Behrens-Fisher problems. The third identity (Section 4), due to Picard (Appell and Kampe de Feriet (1926)), expresses the moments of Savage's (1966) generalization of the Dirichlet distribution as a one-dimensional integral. A generalization of Picard's identity is given. Picard's identity is applied to inference about multinomial cell probabilities, to components-of-variance problems, and to inference from a likelihood function under a Student $t$ distribution of errors.


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James M. Dickey. "Three Multidimensional-integral Identities with Bayesian Applications." Ann. Math. Statist. 39 (5) 1615 - 1628, October, 1968.


Published: October, 1968
First available in Project Euclid: 27 April 2007

zbMATH: 0169.50505
MathSciNet: MR234545
Digital Object Identifier: 10.1214/aoms/1177698143

Rights: Copyright © 1968 Institute of Mathematical Statistics

Vol.39 • No. 5 • October, 1968
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