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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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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)

Subjects:

Q Science > QA Mathematics

Divisions:

Faculty of 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:

Date	Event
2010	Published

Volume:

Vol.2010

Number:

No.7

Number of Pages:

Status:

Not Peer Reviewed

Access rights to Published version:

Open Access

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