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Distribution function estimation by constrained polynomial spline regression

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  • Lan Xue
  • Jing Wang

Abstract

A smooth monotone polynomial spline (PS) estimator is proposed for the cumulative distribution function. The proposed method applies a constrained PS regression to smooth the empirical distribution function, while simultaneously ensures monotonicity by imposing a set of linear constraints on the coefficients of the PS functions. This feature is not shared by its kernel counterpart in [Cheng, M.Y., and Peng, L. (2002), ‘Regression Modeling for Nonparametric Estimation of Distribution and Quantile Functions’, Statistica Sinica, 12, 1043–1060], as the kernel estimator is not necessarily monotone. Under mild assumptions, both L2 and uniform convergence rates are obtained. Our simulation studies show that the proposed estimator has better finite sample performance than the simple empirical distribution function. We also illustrate the use of the proposed method by analysing two real data examples.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:gnstxx:v:22:y:2010:i:4:p:443-457
    DOI: 10.1080/10485250903336802
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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.
    2. Li Wang, 2009. "Single-index model-assisted estimation in survey sampling," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(4), pages 487-504.
    3. Zhou, Jianhui, 2009. "Robust dimension reduction based on canonical correlation," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 195-209, January.
    4. Csörgóo, Miklós & Horváth, Lajos, 1986. "Approximations of weighted empirical and quantile processes," Statistics & Probability Letters, Elsevier, vol. 4(6), pages 275-280, October.
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    Cited by:

    1. 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.
    2. 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).
    3. Im, Jongho & Morikawa, Kosuke & Ha, Hyung-Tae, 2020. "A least squares-type density estimator using a polynomial function," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    4. 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.
    5. Yanlin Tang & Huixia Wang & Xuming He & Zhongyi Zhu, 2012. "An informative subset-based estimator for censored quantile regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(4), pages 635-655, December.

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