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Spline confidence bands for variance functions

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  • Qiongxia Song
  • Lijian Yang

Abstract

Asymptotically exact and conservative confidence bands are obtained for possibly heteroscedastic variance functions, using piecewise constant and piecewise linear spline estimation, respectively. The variance estimation is as efficient as an infeasible estimator when the conditional mean function is known, and the widths of the confidence bands are of optimal order. Simulation experiments provide strong evidence that corroborates the asymptotic theory while the computing is extremely fast. A slower bootstrap band is also proposed, with much higher accuracy. As illustrations, the bootstrap spline band has been applied to test for heteroscedasticity in fossil data and in motorcycle data.

Suggested Citation

  • Qiongxia Song & Lijian Yang, 2009. "Spline confidence bands for variance functions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(5), pages 589-609.
    • Handle: RePEc:taf:gnstxx:v:21:y:2009:i:5:p:589-609
    DOI: 10.1080/10485250902811151
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    Citations

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    Cited by:

    1. 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.
    2. Yanchun Jin, 2016. "Nonparametric tests for the effect of treatment on conditional variance," KIER Working Papers 948, Kyoto University, Institute of Economic Research.
    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. Yujiao Yang & Yuhang Xu & Qiongxia Song, 2012. "Spline confidence bands for variance functions in nonparametric time series regressive models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(3), pages 699-714.
    5. 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.
    6. 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.
    7. 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).
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.

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