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nKernel orthogonal polynomials on the Dirichlet, DirichletMultinomial, PoissonDirichlet and Ewens sampling distributions, and positivedefinite sequences
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Griffiths, Robert C. and Spanò, Dario (2010) nKernel orthogonal polynomials on the Dirichlet, DirichletMultinomial, PoissonDirichlet and Ewens sampling distributions, and positivedefinite 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 socalled 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 DirichletMultinomial distribution, respectively on the continuous and the Ndiscrete ddimensional simplex. Their infinitedimensional limit distributions, respectively the PoissonDirichlet
distribution and the Ewens' sampling formula, are considered as well. We study in particular the
possibility of mapping canonical correlations on the ddimensional continuous simplex (i) to canonical
correlation sequences on the d + 1dimensional 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 northogonal 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 nonuniqueness
di±culties arising when dealing with multivariate orthogonal (or biorthogonal) 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 nontrivial 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 
Volume:  Vol.2010 
Number:  No.7 
Number of Pages:  40 
Status:  Not Peer Reviewed 
Access rights to Published version:  Open Access 
References:  [1] M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and 
URI:  http://wrap.warwick.ac.uk/id/eprint/35071 
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