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A simple bootstrap method for constructing nonparametric confidence bands for functions

Author

Listed:
  • Peter Hall

    (Institute for Fiscal Studies)

  • Joel L. Horowitz

    () (Institute for Fiscal Studies and Northwestern University)

Abstract

Standard approaches to constructing nonparametric confidence bands for functions are frustrated by the impact of bias, which generally is not estimated consistently when using the bootstrap and conventionally smoothed function estimators. To overcome this problem, it is common practice to either undersmooth, so as to reduce the impact of bias, or oversmooth, and thereby introduce an explicit or implicit bias estimator. However, these approaches and others based on nonstandard smoothing methods, complicate the process of inference, for example by requiring the choice of new, unconventional smoothing parameters and, in the case of undersmoothing, producing relatively wide bands. In this paper we suggest a new approach, which exploits to our advantage one of the difficulties that, in the past, has prevented an attractive solution to the problem— the fact that the standard bootstrap bias estimator suffer from relatively high-frequency stochastic error. The high frequency, together with a technique based on quantiles, can be exploited to dampen down the stochastic error term, leading to relatively narrow, simple-to-construct confidence bands. A supplement to this article, which outlines theoretical properties underpinning the methodology and provides a proof of theorem, can be viewed here

Suggested Citation

  • Peter Hall & Joel L. Horowitz, 2013. "A simple bootstrap method for constructing nonparametric confidence bands for functions," CeMMAP working papers CWP29/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:29/13
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    File URL: http://www.cemmap.ac.uk/wps/cwp291313.pdf
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    References listed on IDEAS

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    1. Tiejun Tong & Yuedong Wang, 2005. "Estimating residual variance in nonparametric regression using least squares," Biometrika, Biometrika Trust, vol. 92(4), pages 821-830, December.
    2. Song Xi Chen & Wolfgang Härdle & Ming Li, 2003. "An empirical likelihood goodness-of-fit test for time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(3), pages 663-678.
    3. H rdle, Wolfgang & Huet, Sylvie & Mammen, Enno & Sperlich, Stefan, 2004. "Bootstrap Inference In Semiparametric Generalized Additive Models," Econometric Theory, Cambridge University Press, vol. 20(02), pages 265-300, April.
    4. Horowitz, Joel L & Spokoiny, Vladimir G, 2001. "An Adaptive, Rate-Optimal Test of a Parametric Mean-Regression Model against a Nonparametric Alternative," Econometrica, Econometric Society, vol. 69(3), pages 599-631, May.
    5. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167, April.
    6. Mendez, Guillermo & Lohr, Sharon, 2011. "Estimating residual variance in random forest regression," Computational Statistics & Data Analysis, Elsevier, vol. 55(11), pages 2937-2950, November.
    7. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506, April.
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    Citations

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    Cited by:

    1. Katharina Proksch, 2016. "On confidence bands for multivariate nonparametric regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 209-236, February.
    2. Jean-Pierre Florens & Joel L. Horowitz & Ingred van Keilegom, 2016. "Bias-corrected confidence intervals in a class of linear inverse problems," CeMMAP working papers CWP19/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Mayya Zhilova, 2015. "Simultaneous likelihood-based bootstrap confidence sets for a large number of models," SFB 649 Discussion Papers SFB649DP2015-031, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    4. Horowitz, Joel L. & Lee, Sokbae, 2017. "Nonparametric estimation and inference under shape restrictions," Journal of Econometrics, Elsevier, vol. 201(1), pages 108-126.
    5. Ryo Okui & Takahide Yanagi, 2018. "Kernel Estimation for Panel Data with Heterogeneous Dynamics," Papers 1802.08825, arXiv.org, revised Mar 2018.
    6. Sokbae Lee & Ryo Okui & Yoon-Jae Whang, 2016. "Doubly robust uniform confidence band for the conditional average treatment effect function," CeMMAP working papers CWP03/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Susanne M. Schennach, 2015. "A bias bound approach to nonparametric inference," CeMMAP working papers CWP71/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    8. Joel L. Horowitz & Anand Krishnamurthy, 2017. "A bootstrap method for constructing pointwise and uniform confidence bands for conditional quantile functions," CeMMAP working papers CWP01/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    9. Katharina Proksch, 2016. "On confidence bands for multivariate nonparametric regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 209-236, February.
    10. Timothy B. Armstrong & Michal Kolesár, 2016. "Optimal Inference in a Class of Regression Models," Cowles Foundation Discussion Papers 2043, Cowles Foundation for Research in Economics, Yale University.
    11. Shih-Kang Chao & Katharina Proksch & Holger Dette & Wolfgang Karl Härdle, 2017. "Confidence Corridors for Multivariate Generalized Quantile Regression," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(1), pages 70-85, January.
    12. Matias D. Cattaneo & Max H. Farrell & Yingjie Feng, 2018. "Large Sample Properties of Partitioning-Based Series Estimators," Papers 1804.04916, arXiv.org.
    13. Sebastian Calonico & Matias D. Cattaneo & Max H. Farrell, 2015. "On the Effect of Bias Estimation on Coverage Accuracy in Nonparametric Inference," Papers 1508.02973, arXiv.org, revised Mar 2018.

    More about this item

    Keywords

    bandwidth; bias; boostrap; confidence interval; conservative coverage; coverage error; kernal methods; statistical smoothing;

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