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A Smooth Simultaneous Confidence Corridor for the Mean of Sparse Functional Data

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  • Shuzhuan Zheng
  • Lijian Yang
  • Wolfgang K. Härdle

Abstract

Functional data analysis (FDA) has become an important area of statistics research in the recent decade, yet a smooth simultaneous confidence corridor (SCC) does not exist in the literature for the mean function of sparse functional data. SCC is a powerful tool for making statistical inference on an entire unknown function, nonetheless classic "Hungarian embedding" techniques for establishing asymptotic correctness of SCC completely fail for sparse functional data. We propose a local linear SCC and a shoal of confidence intervals (SCI) for the mean function of sparse functional data, and establish that it is asymptotically equivalent to the SCC of independent regression data, using new results from Gaussian process extreme value theory. The SCC procedure is examined in simulations for its superior theoretical accuracy and performance, and used to analyze growth curve data, confirming findings with quantified high significance levels. Supplementary materials for this article are available online.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:jnlasa:v:109:y:2014:i:506:p:661-673
    DOI: 10.1080/01621459.2013.866899
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    Citations

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    Cited by:

    1. 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).
    2. 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.
    3. 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.
    4. 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).
    5. Yuliana Linke & Igor Borisov & Pavel Ruzankin & Vladimir Kutsenko & Elena Yarovaya & Svetlana Shalnova, 2024. "Multivariate Universal Local Linear Kernel Estimators in Nonparametric Regression: Uniform Consistency," Mathematics, MDPI, vol. 12(12), pages 1-23, June.
    6. Kun Huang & Sijie Zheng & Lijian Yang, 2022. "Inference for dependent error functional data with application to event-related potentials," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(4), pages 1100-1120, December.
    7. 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.
    8. Yuliana Linke & Igor Borisov & Pavel Ruzankin & Vladimir Kutsenko & Elena Yarovaya & Svetlana Shalnova, 2022. "Universal Local Linear Kernel Estimators in Nonparametric Regression," Mathematics, MDPI, vol. 10(15), pages 1-28, July.
    9. Zhou, Ling & Lin, Huazhen & Chen, Kani & Liang, Hua, 2019. "Efficient estimation and computation of parameters and nonparametric functions in generalized semi/non-parametric regression models," Journal of Econometrics, Elsevier, vol. 213(2), pages 593-607.
    10. 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.
    11. 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.
    12. 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.
    13. Hassan Sharghi Ghale-Joogh & S. Mohammad E. Hosseini-Nasab, 2021. "On mean derivative estimation of longitudinal and functional data: from sparse to dense," Statistical Papers, Springer, vol. 62(4), pages 2047-2066, August.
    14. 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.
    15. Jiang, Jiakun & Lin, Huazhen & Zhong, Qingzhi & Li, Yi, 2022. "Analysis of multivariate non-gaussian functional data: A semiparametric latent process approach," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    16. Jialiang Li & Yaguang Li & Tailen Hsing, 2022. "On functional processes with multiple discontinuities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(3), pages 933-972, July.
    17. 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.
    18. 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.
    19. 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.

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