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Calculating bivariate orthonormal polynomials by recurrence

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Abstract
Emerson gave recurrence formulae for the calculation of orthonormal polynomials for univariate discrete random variables. He claimed that as these were based on the Christoffel-Darboux recurrence relation they were more efficient than those based on the Gram-Schmidt method. This approach was generalised by Rayner and colleagues to arbitrary univariate random variables. The only constraint was that the expectations needed are well-defined. Here the approach is extended to arbitrary bivariate random variables for which the expectations needed are well- defined. The extension to multivariate random variables is clear.
Keywords
smooth tests of goodness of fit, Categorical data analysis, copulas, orthonormal polynomials, Emerson polynomials, 2-WAY CONTINGENCY-TABLES, ORTHOGONAL POLYNOMIALS, COPULAS

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MLA
Rayner, John CW, et al. “Calculating Bivariate Orthonormal Polynomials by Recurrence.” AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, vol. 55, no. 1, 2013, pp. 15–24, doi:10.1111/anzs.12011.
APA
Rayner, J. C., Thas, O., Pipelers, P., & Beh, E. J. (2013). Calculating bivariate orthonormal polynomials by recurrence. AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, 55(1), 15–24. https://doi.org/10.1111/anzs.12011
Chicago author-date
Rayner, John CW, Olivier Thas, Peter Pipelers, and Eric J Beh. 2013. “Calculating Bivariate Orthonormal Polynomials by Recurrence.” AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS 55 (1): 15–24. https://doi.org/10.1111/anzs.12011.
Chicago author-date (all authors)
Rayner, John CW, Olivier Thas, Peter Pipelers, and Eric J Beh. 2013. “Calculating Bivariate Orthonormal Polynomials by Recurrence.” AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS 55 (1): 15–24. doi:10.1111/anzs.12011.
Vancouver
1.
Rayner JC, Thas O, Pipelers P, Beh EJ. Calculating bivariate orthonormal polynomials by recurrence. AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS. 2013;55(1):15–24.
IEEE
[1]
J. C. Rayner, O. Thas, P. Pipelers, and E. J. Beh, “Calculating bivariate orthonormal polynomials by recurrence,” AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, vol. 55, no. 1, pp. 15–24, 2013.
@article{2043184,
  abstract     = {{Emerson gave recurrence formulae for the calculation of orthonormal polynomials for univariate discrete random variables. He claimed that as these were based on the Christoffel-Darboux recurrence relation they were more efficient than those based on the Gram-Schmidt method. This approach was generalised by Rayner and colleagues to arbitrary univariate random variables. The only constraint was that the expectations needed are well-defined. Here the approach is extended to arbitrary bivariate random variables for which the expectations needed are well- defined. The extension to multivariate random variables is clear.}},
  author       = {{Rayner, John CW and Thas, Olivier and Pipelers, Peter and Beh, Eric J}},
  issn         = {{1369-1473}},
  journal      = {{AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS}},
  keywords     = {{smooth tests of goodness of fit,Categorical data analysis,copulas,orthonormal polynomials,Emerson polynomials,2-WAY CONTINGENCY-TABLES,ORTHOGONAL POLYNOMIALS,COPULAS}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{15--24}},
  title        = {{Calculating bivariate orthonormal polynomials by recurrence}},
  url          = {{http://doi.org/10.1111/anzs.12011}},
  volume       = {{55}},
  year         = {{2013}},
}

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