Three Multidimensional-integral Identities with Bayesian Applications

@article{Dickey68, abstract = {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.}, author = {Dickey, James M. }, citeulike-article-id = {2898859}, journal = {The Annals of Mathematical Statistics}, keywords = {file-import-08-06-16}, number = {5}, pages = {1615--1628}, posted-at = {2008-06-16 16:10:56}, priority = {2}, publisher = {Institute of Mathematical Statistics}, title = {Three Multidimensional-integral Identities with Bayesian Applications}, url = {http://www.jstor.org/stable/2239419}, volume = {39}, year = {1968} }

TY - JOUR ID - Dickey68 L3 - citeulike-article-id:2898859 TI - Three Multidimensional-integral Identities with Bayesian Applications JF - The Annals of Mathematical Statistics VL - 39 IS - 5 SP - 1615 EP - 1628 PB - Institute of Mathematical Statistics N2 - 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. KW - file-import-08-06-16 AU - Dickey, James PY - 1968/// UR - http://www.jstor.org/stable/2239419 ER -