A Generalisation of Emerson’s recurrence formulae and the Gray-Williams index

Eric J. Beh () and Rosaria Lombardo ()
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Eric J. Beh: University of Wollongong
Rosaria Lombardo: University of Campania “Luigi Vanvitelli”

Metrika: International Journal for Theoretical and Applied Statistics, 2025, vol. 88, issue 6, No 10, 1005-1022

Abstract: Abstract When analysing the association between the ordered categorical variables of a contingency table, orthogonal polynomials derived from the recurrence formulae of Emerson (1968, Biometrics, 24: 695 - 701) have been extensively used. The calculation of such polynomials is somewhat limited because they reflect only the univariate structure of each variable. This paper proposes a new generalisation of Emerson’s recurrence formulae that reflects bivariate and, more generally, multivariate, association structures for the construction of orthogonal polynomials for multi-way contingency tables. We shall demonstrate the utility of this generalisation by giving special attention to the Gray-Williams index, a three-way variant of the Goodman-Kruskal tau index.

Keywords: Gray-Williams index; Linear; quadratic and higher order components; Univariate orthogonal polynomials (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s00184-024-00981-1

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