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On the smoothness of conditional expectation functionals

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  • Song, Kyungchul

Abstract

Given a class Λ of real functions and a twice differentiable real-valued map φ on R, let Γ be an R-valued functional on Λ of form Γ:λ↦E[Z⋅φ(E[Y∣λ(X)])], where Z and Y are random variables and X is a random vector. This paper calls Γ a conditional expectation functional. Conditional expectation functionals often arise in semiparametric models. The main contribution of this paper is that it provides nontrivial conditions under which Γ has a uniform modulus of continuity with order 2. Hence under these conditions, the functional Γ becomes very smooth.

Suggested Citation

  • Song, Kyungchul, 2012. "On the smoothness of conditional expectation functionals," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 1028-1034.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:5:p:1028-1034
    DOI: 10.1016/j.spl.2012.01.027
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    Cited by:

    1. Mammen, Enno & Rothe, Christoph & Schienle, Melanie, 2016. "Semiparametric Estimation With Generated Covariates," Econometric Theory, Cambridge University Press, vol. 32(5), pages 1140-1177, October.
    2. Ying-Ying Lee, 2018. "Partial Mean Processes with Generated Regressors: Continuous Treatment Effects and Nonseparable Models," Papers 1811.00157, arXiv.org.
    3. Hisatoshi Tanaka, 2020. "Differentiability of the Conditional Expectation," Working Papers 1920, Waseda University, Faculty of Political Science and Economics.
    4. Song, Kyungchul, 2014. "Semiparametric models with single-index nuisance parameters," Journal of Econometrics, Elsevier, vol. 178(P3), pages 471-483.
    5. Ying-Ying Lee, 2014. "Partial Mean Processes with Generated Regressors: Continuous Treatment Effects and Nonseparable Models," Economics Series Working Papers 706, University of Oxford, Department of Economics.

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