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Kernel estimation of multivariate cumulative distribution function

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  • Rong Liu
  • Lijian Yang

Abstract

A smooth kernel estimator is proposed for multivariate cumulative distribution functions (cdf), extending the work of Yamato [H. Yamato, Uniform convergence of an estimator of a distribution function, Bull. Math. Statist. 15 (1973), pp. 69–78.] on univariate distribution function estimation. Under assumptions of strict stationarity and geometrically strong mixing, we establish that the proposed estimator follows the same pointwise asymptotically normal distribution of the empirical cdf, while the new estimator is a smooth instead of a step function as the empirical cdf. We also show that under stronger assumptions the smooth kernel estimator has asymptotically smaller mean integrated squared error than the empirical cdf, and converges to the true cdf uniformly almost surely at a rate of (n−1/2log n). Simulated examples are provided to illustrate the theoretical properties. Using the smooth estimator, survival curves for US gross domestic product (GDP) growth are estimated conditional on the unemployment growth rate to examine how GDP growth rate depends on the unemployment policy. Another example of gold and silver price returns is given.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:gnstxx:v:20:y:2008:i:8:p:661-677
    DOI: 10.1080/10485250802326391
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    References listed on IDEAS

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    1. L. Yang & R. Tschernig, 1999. "Multivariate bandwidth selection for local linear regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(4), pages 793-815.
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    1. Fousekis, Panos & Grigoriadis, Vasilis, 2017. "Price co-movement and the crack spread in the US futures markets," Journal of Commodity Markets, Elsevier, vol. 7(C), pages 57-71.
    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. Alexandre Leblanc, 2012. "On estimating distribution functions using Bernstein polynomials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(5), pages 919-943, October.
    4. Arup Bose & Santanu Dutta, 2022. "Kernel based estimation of the distribution function for length biased data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(3), pages 269-287, April.
    5. Li, Genyuan & Rabitz, Herschel, 2017. "Relationship between sensitivity indices defined by variance- and covariance-based methods," Reliability Engineering and System Safety, Elsevier, vol. 167(C), pages 136-157.
    6. Catalina Bolancé & Carlos Alberto Acuña, 2021. "A New Kernel Estimator of Copulas Based on Beta Quantile Transformations," Mathematics, MDPI, vol. 9(10), pages 1-16, May.
    7. Funke, Benedikt & Palmes, Christian, 2017. "A note on estimating cumulative distribution functions by the use of convolution power kernels," Statistics & Probability Letters, Elsevier, vol. 121(C), pages 90-98.
    8. Sylvain Chassang & Kei Kawai & Jun Nakabayashi & Juan Ortner, 2019. "Data Driven Regulation: Theory and Application to Missing Bids," Boston University - Department of Economics - Working Papers Series WP2019-04, Boston University - Department of Economics.
    9. Jeffrey Racine, 2015. "Mixed data kernel copulas," Empirical Economics, Springer, vol. 48(1), pages 37-59, February.
    10. 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.
    11. Langrené, Nicolas & Warin, Xavier, 2021. "Fast multivariate empirical cumulative distribution function with connection to kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 162(C).
    12. 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.
    13. 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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