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On a Simple Identity for the Conditional Expectation of Orthogonal Polynomials


Consider a two-dimensional random vector (X, Y )T. Let Q0, Q1,… denote orthogonal polynomials with respect to the marginal distribution of X and let P0, P1,… denote orthogonal polynomials with respect to the marginal distribution of Y. In this paper, identities of the form E[Pn(Y )|X] = anQn(X), for constants a0, a1,… are considered and necessary and sufficient conditions for this type of identity to hold are given,. The application of the identity to the maximal correlation of two random variables and to the L2 completeness of a bivariate distribution are discussed.

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  • Andrews, D.W.K. (2017). Examples of L2-complete and boundedly-complete distributions. J. Econ.199, 213–220.

    Article  Google Scholar 

  • Andrews, G.E., Askey, R. and Roy, R. (1999). Special Functions. Cambridge University Press, Cambridge.

    Book  Google Scholar 

  • Gebelein, H. (1941). Das statistische Problem der Korrelation als Variations und Eigenwertproblem und sein Zusammenhang mit der Ausgleichsrechnung. Z. Angew. Math. Mech.21, 364–379.

    MathSciNet  Article  Google Scholar 

  • Granger, C.W.J. and Newbold, P. (1976). Forecasting transformed series. J. Roy Statist. Soc. B38, 189–203.

    MathSciNet  MATH  Google Scholar 

  • Johnson, N.L. and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York.

    MATH  Google Scholar 

  • Lancaster, H.O. (1957). Some properties of the bivariate normal distribution considered in the form of a contingency table. Biometrika44, 189–203.

    Article  Google Scholar 

  • Lancaster, H.O. (1958). The structure of bivariate distributions. Ann. Math. Stat.29, 719–736.

    MathSciNet  Article  Google Scholar 

  • Patel, J.K. and Read, C.B. (1996). Handbook of the Normal Distribution, 2nd edn. Marcel Dekker, New York.

    MATH  Google Scholar 

  • Severini, T.A. and Tripathi, G. (2006). Some identification issues in nonparametric linear models with endogenous regressors. Economet. Theor.22, 258–278.

    MathSciNet  Article  Google Scholar 

  • Szegö, G (1975). Orthogonal Polynomials. American Mathematical Society, Providence.

    MATH  Google Scholar 

  • Temme, N.M. (1996). Special Functions. Wiley, New York.

    Book  Google Scholar 

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I would like to thank the associate editor and referees for a number of useful comments.

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Correspondence to Thomas A. Severini.

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Severini, T.A. On a Simple Identity for the Conditional Expectation of Orthogonal Polynomials. Sankhya A 82, 13–27 (2020).

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  • Bivariate Dirichlet distribution
  • Bivariate gamma distribution
  • Jacobi polynomials
  • L2 completeness
  • Maximal correlation
  • Mehler’s identity

AMS (2000) subject classification.

  • Primary 42C05
  • Secondary 60E05