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On confidence bands in nonparametric density estimation and regression

Author

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  • Hall, Peter
  • Titterington, D. M.

Abstract

We describe a unified approach to the construction of confidence bands in nonparametric density estimation and regression. Our techniques are based on interpolation formulae in numerical differentiation, and our arguments generate a variety of bands depending on the assumptions one is prepared to make about derivatives of the unknown function. The bands are simultaneous, in the sense that they contain the entire function with probability at least an amount. The order of magnitude of the minimum width of any confidence band is described, and our bands are shown to achieve that order. Examples illustrate applications of the technique.

Suggested Citation

  • Hall, Peter & Titterington, D. M., 1988. "On confidence bands in nonparametric density estimation and regression," Journal of Multivariate Analysis, Elsevier, vol. 27(1), pages 228-254, October.
    • Handle: RePEc:eee:jmvana:v:27:y:1988:i:1:p:228-254
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    Citations

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    Cited by:

    1. Manuel Wiesenfarth & Carlos Matías Hisgen & Thomas Kneib & Carmen Cadarso-Suarez, 2014. "Bayesian Nonparametric Instrumental Variables Regression Based on Penalized Splines and Dirichlet Process Mixtures," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(3), pages 468-482, July.
    2. Horowitz, Joel L. & Lee, Sokbae, 2012. "Uniform confidence bands for functions estimated nonparametrically with instrumental variables," Journal of Econometrics, Elsevier, vol. 168(2), pages 175-188.
    3. Lei Gao & Li Wang, 2011. "Security price responses to unexpected earnings: a nonparametric investigation," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 20(2), pages 241-258, June.
    4. J. Cristóbal & J. Ojeda & J. Alcalá, 2004. "Confidence bands in nonparametric regression with length biased data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(3), pages 475-496, September.
    5. Schüssler, Rainer & Trede, Mark, 2016. "Constructing minimum-width confidence bands," Economics Letters, Elsevier, vol. 145(C), pages 182-185.
    6. Jing Wang & Lijian Yang, 2009. "Efficient and fast spline-backfitted kernel smoothing of additive models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(3), pages 663-690, September.
    7. Tribouley, Karine, 2004. "Adaptive simultaneous confidence intervals in non-parametric estimation," Statistics & Probability Letters, Elsevier, vol. 69(1), pages 37-51, August.
    8. Weichi Wu & Zhou Zhou, 2017. "Nonparametric Inference for Time-Varying Coefficient Quantile Regression," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(1), pages 98-109, January.
    9. Li Cai & Lijian Yang, 2015. "A smooth simultaneous confidence band for conditional variance function," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 632-655, September.
    10. Lijie Gu & Li Wang & Wolfgang Härdle & Lijian Yang, 2014. "A simultaneous confidence corridor for varying coefficient regression with sparse functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(4), pages 806-843, December.
    11. Xiang Zhang & Yanbing Zheng, 2014. "Nonparametric Bayesian inference for multivariate density functions using Feller priors," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(2), pages 321-340, June.
    12. Colin O. Wu & Kai F. Yu, 2002. "Nonparametric Varying-Coefficient Models for the Analysis of Longitudinal Data," International Statistical Review, International Statistical Institute, vol. 70(3), pages 373-393, December.
    13. Peter Hall & Joel L. Horowitz, 2012. "A simple bootstrap method for constructing nonparametric confidence bands for functions," CeMMAP working papers 14/12, Institute for Fiscal Studies.
    14. Cheng, Ming-Yen & Hall, Peter, 1998. "On mode testing and empirical approximations to distributions," Statistics & Probability Letters, Elsevier, vol. 39(3), pages 245-254, August.
    15. Peter B. Gilbert & L. J. Wei & Michael R. Kosorok & John D. Clemens, 2002. "Simultaneous Inferences on the Contrast of Two Hazard Functions with Censored Observations," Biometrics, The International Biometric Society, vol. 58(4), pages 773-780, December.
    16. Liugen Xue, 2010. "Empirical Likelihood Local Polynomial Regression Analysis of Clustered Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(4), pages 644-663, December.
    17. Li Cai & Lijie Gu & Qihua Wang & Suojin Wang, 2021. "Simultaneous confidence bands for nonparametric regression with missing covariate data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(6), pages 1249-1279, December.
    18. K. De Brabanter & Y. Liu & C. Hua, 2016. "Convergence rates for uniform confidence intervals based on local polynomial regression estimators," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(1), pages 31-48, March.
    19. Xue, Liugen & Xue, Dong, 2011. "Empirical likelihood for semiparametric regression model with missing response data," Journal of Multivariate Analysis, Elsevier, vol. 102(4), pages 723-740, April.
    20. Peter Hall & Joel L. Horowitz, 2013. "A simple bootstrap method for constructing nonparametric confidence bands for functions," CeMMAP working papers 29/13, Institute for Fiscal Studies.
    21. Li Cai & Suojin Wang, 2021. "Global statistical inference for the difference between two regression mean curves with covariates possibly partially missing," Statistical Papers, Springer, vol. 62(6), pages 2573-2602, December.
    22. Yun Fang & Li-Xing Zhu, 2012. "Asymptotics of SIMEX-based variance estimation," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(3), pages 329-345, April.
    23. Yuanyuan Zhang & Lijian Yang, 2018. "A smooth simultaneous confidence band for correlation curve," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(2), pages 247-269, June.
    24. Chen Zhong & Lijian Yang, 2021. "Simultaneous confidence bands for comparing variance functions of two samples based on deterministic designs," Computational Statistics, Springer, vol. 36(2), pages 1197-1218, June.
    25. M. P. Wand & J. C. F. Yu, 2022. "Density estimation via Bayesian inference engines," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(2), pages 199-216, June.

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