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On a Simple Identity for the Conditional Expectation of Orthogonal Polynomials

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  • Thomas A. Severini

    (Northwestern University)

Abstract

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.

Suggested Citation

  • Thomas A. Severini, 2020. "On a Simple Identity for the Conditional Expectation of Orthogonal Polynomials," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(1), pages 13-27, February.
    • Handle: RePEc:spr:sankha:v:82:y:2020:i:1:d:10.1007_s13171-018-00161-0
    DOI: 10.1007/s13171-018-00161-0
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    References listed on IDEAS

    as
    1. Severini, Thomas A. & Tripathi, Gautam, 2006. "Some Identification Issues In Nonparametric Linear Models With Endogenous Regressors," Econometric Theory, Cambridge University Press, vol. 22(2), pages 258-278, April.
    2. Andrews, Donald W.K., 2017. "Examples of L2-complete and boundedly-complete distributions," Journal of Econometrics, Elsevier, vol. 199(2), pages 213-220.
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    More about this item

    Keywords

    Bivariate Dirichlet distribution; Bivariate gamma distribution; Jacobi polynomials; L2 completeness; Maximal correlation; Mehler’s identity;
    All these keywords.

    JEL classification:

    • L2 - Industrial Organization - - Firm Objectives, Organization, and Behavior

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