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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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Citation

Please use this url to cite or link to this publication:

MLA
Rayner, John CW, Olivier Thas, Peter Pipelers, et al. “Calculating Bivariate Orthonormal Polynomials by Recurrence.” AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS 55.1 (2013): 15–24. Print.
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.
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.
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.
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://dx.doi.org/10.1111/anzs.12011},
  volume       = {55},
  year         = {2013},
}

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