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n-Kernel orthogonal polynomials on the Dirichlet, Dirichlet-Multinomial, Poisson-Dirichlet and Ewens sampling distributions, and positive-definite sequences
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Griffiths, Robert C. and Spanò, Dario (2010) n-Kernel orthogonal polynomials on the Dirichlet, Dirichlet-Multinomial, Poisson-Dirichlet and Ewens sampling distributions, and positive-definite sequences. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. Working papers, Vol.2010 (No.7).
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Official URL: http://www2.warwick.ac.uk/fac/sci/statistics/crism...
Abstract
We consider a multivariate version of the so-called Lancaster problem of characterizing
canonical correlation coe±cients of symmetric bivariate distributions with identical marginals and
orthogonal polynomial expansions. The marginal distributions examined in this paper are the Dirichlet and the Dirichlet-Multinomial distribution, respectively on the continuous and the N-discrete d-dimensional simplex. Their infinite-dimensional limit distributions, respectively the Poisson-Dirichlet
distribution and the Ewens' sampling formula, are considered as well. We study in particular the
possibility of mapping canonical correlations on the d-dimensional continuous simplex (i) to canonical
correlation sequences on the d + 1-dimensional simplex and/or (ii) to canonical correlations on the
discrete simplex, and viceversa. Driven by this motivation, the first half of the paper is devoted to
providing a full characterization and probabilistic interpretation of |n|-orthogonal polynomial kernels
(i.e. sums of products of orthogonal polynomials of the same degree |n|) with respect to the mentioned
marginal distributions. Orthogonal polynomial kernels are important to overcome some non-uniqueness
di±culties arising when dealing with multivariate orthogonal (or bi-orthogonal) polynomials.We estab-
lish several identities and some integral representations which are multivariate extensions of important
results known for the case d = 2 since the 1970's. These results, along with a common interpretation
of the mentioned kernels in terms of dependent Polya urns, are shown to be key features leading to
several non-trivial solutions to Lancaster's problem, many of which can be extended naturally to the
limit as d -> ∞.
Item Type: | Working or Discussion Paper (Working Paper) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||
Library of Congress Subject Headings (LCSH): | Multivariate analysis, Correlation (Statistics), Marginal distributions, Kernel functions | ||||
Series Name: | Working papers | ||||
Publisher: | University of Warwick. Centre for Research in Statistical Methodology | ||||
Place of Publication: | Coventry | ||||
Official Date: | 2010 | ||||
Dates: |
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Volume: | Vol.2010 | ||||
Number: | No.7 | ||||
Number of Pages: | 40 | ||||
Status: | Not Peer Reviewed | ||||
Access rights to Published version: | Open Access (Creative Commons) | ||||
Date of first compliant deposit: | 1 August 2016 | ||||
Date of first compliant Open Access: | 1 August 2016 |
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