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April, 1967 Expansions of $t$ Densities and Related Complete Integrals
James M. Dickey
Ann. Math. Statist. 38(2): 503-510 (April, 1967). DOI: 10.1214/aoms/1177698966


A class of alternatives is here presented to Fisher's (1925) expansion of Student's $t$ density function. These expansions involve Appell's polynomials; and hence, recurrence schemes are available for the coefficients. Complete integrals of products of $t$ densities are of interest as Behrens-Fisher densities (viewed as Bayesian posterior distributions: Jeffreys, 1940; Patil, 1964) as moments of Bayesian posterior distributions (Anscombe, 1963; Tiao and Zellner, 1964). A symptotic expansion of complete integrals, obtained by term-by-term integration of these expansions, are favorably compared with those obtained from Fisher's expansion. Although expansions of complete integrals of products of multivariate $t$ densities can be developed from these expansions by the methods of Tiao and Zellner, the resulting coefficients are practically as complicated as the Tiao and Zellner coefficients; methods will be published soon (Dickey, 1965) for reducing the dimensionality of such integrals for quadrature. The paper concludes with a numerical study of the integral expansions.


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James M. Dickey. "Expansions of $t$ Densities and Related Complete Integrals." Ann. Math. Statist. 38 (2) 503 - 510, April, 1967.


Published: April, 1967
First available in Project Euclid: 27 April 2007

zbMATH: 0168.17901
MathSciNet: MR208711
Digital Object Identifier: 10.1214/aoms/1177698966

Rights: Copyright © 1967 Institute of Mathematical Statistics

Vol.38 • No. 2 • April, 1967
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