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Uniform Convergence Results for the Local Linear Regression Estimation of the Conditional Distribution

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  • Haitian Xie

Abstract

This paper examines the local linear regression (LLR) estimate of the conditional distribution function $F(y|x)$. We derive three uniform convergence results: the uniform bias expansion, the uniform convergence rate, and the uniform asymptotic linear representation. The uniformity in the above results is with respect to both $x$ and $y$ and therefore has not previously been addressed in the literature on local polynomial regression. Such uniform convergence results are especially useful when the conditional distribution estimator is the first stage of a semiparametric estimator. We demonstrate the usefulness of these uniform results with two examples: the stochastic equicontinuity condition in $y$, and the estimation of the integrated conditional distribution function.

Suggested Citation

  • Haitian Xie, 2021. "Uniform Convergence Results for the Local Linear Regression Estimation of the Conditional Distribution," Papers 2112.08546, arXiv.org, revised Jun 2023.
  • Handle: RePEc:arx:papers:2112.08546
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    References listed on IDEAS

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    1. Srinjoy Das & Dimitris N. Politis, 2020. "Nonparametric Estimation of the Conditional Distribution at Regression Boundary Points," The American Statistician, Taylor & Francis Journals, vol. 74(3), pages 233-242, July.
    2. Hansen, Bruce E., 2008. "Uniform Convergence Rates For Kernel Estimation With Dependent Data," Econometric Theory, Cambridge University Press, vol. 24(3), pages 726-748, June.
    3. Hall, Peter & Wolff, Rodney C. L. & Yao, Qiwei, 1999. "Methods for estimating a conditional distribution function," LSE Research Online Documents on Economics 6631, London School of Economics and Political Science, LSE Library.
    4. Yangin Fan & Emmanuel Guerre, 2016. "Multivariate Local Polynomial Estimators: Uniform Boundary Properties and Asymptotic Linear Representation," Advances in Econometrics, in: Essays in Honor of Aman Ullah, volume 36, pages 489-537, Emerald Group Publishing Limited.
    5. Kong, Efang & Linton, Oliver & Xia, Yingcun, 2010. "Uniform Bahadur Representation For Local Polynomial Estimates Of M-Regression And Its Application To The Additive Model," Econometric Theory, Cambridge University Press, vol. 26(5), pages 1529-1564, October.
    6. Elias Masry, 1996. "Multivariate Local Polynomial Regression For Time Series:Uniform Strong Consistency And Rates," Journal of Time Series Analysis, Wiley Blackwell, vol. 17(6), pages 571-599, November.
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    Cited by:

    1. Haitian Xie, 2022. "Nonlinear and Nonseparable Structural Functions in Fuzzy Regression Discontinuity Designs," Papers 2204.08168, arXiv.org, revised Jul 2022.
    2. Xie, Haitian, 2024. "Nonlinear and nonseparable structural functions in regression discontinuity designs with a continuous treatment," Journal of Econometrics, Elsevier, vol. 242(1).

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