IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v155y2021ics0167947320301973.html
   My bibliography  Save this article

Smooth simultaneous confidence band for the error distribution function in nonparametric regression

Author

Listed:
  • Gu, Lijie
  • Wang, Suojin
  • Yang, Lijian

Abstract

A smooth simultaneous confidence band (SCB) is constructed for the distribution of unobserved errors in a nonparametric regression model based on a plug-in kernel distribution estimator. The normalized estimation error process is shown to converge to a Gaussian process. Simulation experiments indicate that the proposed SCB not only strikes an intelligent balance between coverage probability and precision, but also achieves surprisingly as much as double efficiency of the classical infeasible SCB. Furthermore, extensive empirical studies are carried out to compare the proposed method with the smooth residual bootstrap method in order to demonstrate the usefulness of each of these methods. As an illustration, the proposed SCB is applied to the Old Faithful geyser data for testing the error distribution.

Suggested Citation

  • Gu, Lijie & Wang, Suojin & Yang, Lijian, 2021. "Smooth simultaneous confidence band for the error distribution function in nonparametric regression," Computational Statistics & Data Analysis, Elsevier, vol. 155(C).
    • Handle: RePEc:eee:csdana:v:155:y:2021:i:c:s0167947320301973
    DOI: 10.1016/j.csda.2020.107106
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167947320301973
    Download Restriction: Full text for ScienceDirect subscribers only.
    File URL: https://libkey.io/10.1016/j.csda.2020.107106?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Krivobokova, Tatyana & Kneib, Thomas & Claeskens, Gerda, 2010. "Simultaneous Confidence Bands for Penalized Spline Estimators," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 852-863.
    2. Michael G. Akritas & Ingrid Van Keilegom, 2001. "Non‐parametric Estimation of the Residual Distribution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(3), pages 549-567, September.
    3. Holger Dette & Natalie Neumeyer & Ingrid Van Keilegom, 2007. "A new test for the parametric form of the variance function in non‐parametric regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(5), pages 903-917, November.
    4. Cheng, Fuxia, 2002. "Consistency of error density and distribution function estimators in nonparametric regression," Statistics & Probability Letters, Elsevier, vol. 59(3), pages 257-270, October.
    5. Y. Xia, 1998. "Bias‐corrected confidence bands in nonparametric regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(4), pages 797-811.
    6. Jiangyan Wang & Suojin Wang & Lijian Yang, 2016. "Simultaneous confidence bands for the distribution function of a finite population and of its superpopulation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(4), pages 692-709, December.
    7. Shuzhuan Zheng & Rong Liu & Lijian Yang & Wolfgang K. Härdle, 2016. "Statistical inference for generalized additive models: simultaneous confidence corridors and variable selection," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(4), pages 607-626, December.
    8. 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.
    9. 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.
    10. Lan Xue & Jing Wang, 2010. "Distribution function estimation by constrained polynomial spline regression," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(4), pages 443-457.
    11. 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.
    12. Lijie Gu & Suojin Wang & Lijian Yang, 2019. "Simultaneous confidence bands for the distribution function of a finite population in stratified sampling," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(4), pages 983-1005, August.
    13. Jiangyan Wang & Fuxia Cheng & Lijian Yang, 2013. "Smooth simultaneous confidence bands for cumulative distribution functions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 25(2), pages 395-407, June.
    14. Natalie Neumeyer, 2009. "Smooth Residual Bootstrap for Empirical Processes of Non‐parametric Regression Residuals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 204-228, June.
    15. Guanqun Cao & Lijian Yang & David Todem, 2012. "Simultaneous inference for the mean function based on dense functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(2), pages 359-377.
    16. Shuzhuan Zheng & Lijian Yang & Wolfgang K. Härdle, 2014. "A Smooth Simultaneous Confidence Corridor for the Mean of Sparse Functional Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(506), pages 661-673, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Majid Mojirsheibani, 2022. "On the maximal deviation of kernel regression estimators with NMAR response variables," Statistical Papers, Springer, vol. 63(5), pages 1677-1705, October.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wang, Jiangyan & Gu, Lijie & Yang, Lijian, 2022. "Oracle-efficient estimation for functional data error distribution with simultaneous confidence band," Computational Statistics & Data Analysis, Elsevier, vol. 167(C).
    2. 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.
    3. Jie Li & Jiangyan Wang & Lijian Yang, 2022. "Kolmogorov–Smirnov simultaneous confidence bands for time series distribution function," Computational Statistics, Springer, vol. 37(3), pages 1015-1039, July.
    4. 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.
    5. Lijie Gu & Suojin Wang & Lijian Yang, 2019. "Simultaneous confidence bands for the distribution function of a finite population in stratified sampling," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(4), pages 983-1005, August.
    6. Jiangyan Wang & Suojin Wang & Lijian Yang, 2016. "Simultaneous confidence bands for the distribution function of a finite population and of its superpopulation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(4), pages 692-709, December.
    7. Li Cai & Lisha Li & Simin Huang & Liang Ma & Lijian Yang, 2020. "Oracally efficient estimation for dense functional data with holiday effects," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(1), pages 282-306, March.
    8. 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.
    9. Feve, Frederique & Florens, Jean-Pierre & Van Keilegom, Ingrid, 2012. "Estimation of conditional ranks and tests of exogeneity in nonparametric nonseparable models," LIDAM Discussion Papers ISBA 2012036, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    10. Holger Dette & Juan Carlos Pardo‐Fernández & Ingrid Van Keilegom, 2009. "Goodness‐of‐Fit Tests for Multiplicative Models with Dependent Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(4), pages 782-799, December.
    11. Natalie Neumeyer, 2009. "Smooth Residual Bootstrap for Empirical Processes of Non‐parametric Regression Residuals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(2), pages 204-228, June.
    12. Jun Zhang & Zhenghui Feng & Xiaoguang Wang, 2018. "A constructive hypothesis test for the single-index models with two groups," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(5), pages 1077-1114, October.
    13. Neumeyer, Natalie & Van Keilegom, Ingrid, 2010. "Estimating the error distribution in nonparametric multiple regression with applications to model testing," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1067-1078, May.
    14. Florens, Jean-Pierre & Simar, Léopold & Van Keilegom, Ingrid, 2014. "Frontier estimation in nonparametric location-scale models," Journal of Econometrics, Elsevier, vol. 178(P3), pages 456-470.
    15. Cao, Guanqun & Wang, Li, 2018. "Simultaneous inference for the mean of repeated functional data," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 279-295.
    16. Wenceslao González-Manteiga & Rosa Crujeiras, 2013. "An updated review of Goodness-of-Fit tests for regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 361-411, September.
    17. Zhong, Chen, 2024. "Oracle-efficient estimation and trend inference in non-stationary time series with trend and heteroscedastic ARMA error," Computational Statistics & Data Analysis, Elsevier, vol. 193(C).
    18. Kiwitt, Sebastian & Nagel, Eva-Renate & Neumeyer, Natalie, 2005. "Empirical likelihood estimators for the error distribution in nonparametric regression models," Technical Reports 2005,45, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    19. Yueying Wang & Guannan Wang & Li Wang & R. Todd Ogden, 2020. "Simultaneous confidence corridors for mean functions in functional data analysis of imaging data," Biometrics, The International Biometric Society, vol. 76(2), pages 427-437, June.
    20. Shuzhuan Zheng & Rong Liu & Lijian Yang & Wolfgang K. Härdle, 2016. "Statistical inference for generalized additive models: simultaneous confidence corridors and variable selection," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(4), pages 607-626, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:csdana:v:155:y:2021:i:c:s0167947320301973. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/csda .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.