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On the smoothness of conditional expectation functionals


  • Song, Kyungchul


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. Song, Kyungchul, 2014. "Semiparametric models with single-index nuisance parameters," Journal of Econometrics, Elsevier, vol. 178(P3), pages 471-483.
    2. 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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