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Simultaneous confidence bands for the distribution function of a finite population and of its superpopulation

Author

Listed:
  • Jiangyan Wang

    (Soochow University)

  • Suojin Wang

    (Texas A&M University)

  • Lijian Yang

    (Tsinghua University)

Abstract

Simultaneous confidence bands (SCBs) are proposed for the distribution function of a finite population and of the latent superpopulation via the empirical distribution function (nonsmooth) and kernel distribution estimator (smooth) based on a simple random sample (SRS), either with or without finite population correction. It is shown that both nonsmooth and smooth SCBs achieve asymptotically the nominal confidence level under standard assumptions. In particular, the uncorrected nonsmooth SCB for superpopulation is exactly the same as the Kolmogorov–Smirnov SCB based on an independent and identically distributed sample as long as the SRS size is infinitesimal relative to the finite population size. Extensive simulation studies confirm the asymptotic properties. As an illustration, the proposed SCBs are constructed for the population distribution of the well-known baseball data (Lohr, Sampling: design and analysis, 2nd edn. Brooks/Cole, Boston, 2009).

Suggested Citation

  • 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.
    • Handle: RePEc:spr:testjl:v:25:y:2016:i:4:d:10.1007_s11749-016-0491-5
    DOI: 10.1007/s11749-016-0491-5
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    References listed on IDEAS

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    1. Rong Liu & Lijian Yang, 2008. "Kernel estimation of multivariate cumulative distribution function," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(8), pages 661-677.
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    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
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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. 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.
    3. 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.
    4. 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.
    5. 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).
    6. 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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