Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Nonparametric Estimation of Distribution Function using Bezier Curve

  • Bae, Whasoo (Department of Data Science/Institute of Statistical Information, Inje University) ;
  • Kim, Ryeongah (Department of Statistics, Pusan National University) ;
  • Kim, Choongrak (Department of Statistics, Pusan National University)
  • Received : 2013.12.03
  • Accepted : 2014.01.16
  • Published : 2014.01.31
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Abstract

In this paper we suggest an efficient method to estimate the distribution function using the Bezier curve, and compare it with existing methods by simulation studies. In addition, we suggest a robust version of cross-validation criterion to estimate the number of Bezier points, and showed that the proposed method is better than the existing methods based on simulation studies.

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References

  1. Bae, W., Choi, H., Park, B.-U. and Kim, C. (2005). Smoothing techniques for the bivariate Kaplan-Meier estimator, Communications in Statistics - Theory and Methods, 34, 1659-1674. https://doi.org/10.1081/STA-200063317
  2. Bezier, P. (1977). Essay de definition numerique des courbes et des surfaces experimentals. Ph.D. dissertation, University of Paris VI.
  3. Bowman, A. W. (1984). An alternative method of cross-validation for the smoothing of density estimates, Biometrika, 71, 353-360. https://doi.org/10.1093/biomet/71.2.353
  4. Bowman, A.W., Hall, P. and Titterington, D. M. (1984). Cross-validation in nonparametric estimation of probabilities and probability densities, Biometrika, 71, 341-351. https://doi.org/10.1093/biomet/71.2.341
  5. Eubank, R. L. (1988). Spline Smoothing and Nonparametric Regression, Marcel Dekker, New York.
  6. Farin, G. E. (1990). Curves and Surfaces for Computer Aided Geometric Design, Academic Press, London.
  7. Geisser, S. (1975). The predictive sample reuse method with applications, Journal of the American Statistical Association, 70, 320-328. https://doi.org/10.1080/01621459.1975.10479865
  8. Hall, P. (1983). Large sample optimality of least squares cross-validation in density estimation, The Annals of Statistics, 4, 1156-1174.
  9. Kim, C. (1996). Nonparametric density estimation via the Bezier curve, Proceedings of the Section on Statistical Graphics of the American Statistical Association, 25-28.
  10. Kim, C., Hong, C. and Jeong, M. (2000). Simulation-Extrapolation via the Bezier curve in measurement error models, Communications in Statistics - Simulation and Computation, 29, 1135-1147. https://doi.org/10.1080/03610910008813657
  11. Kim, C., Kim, W., Hong, C., Park, B.-U. and Jeong, M. (1999). Smoothing techniques via the Bezier curve, Communications in Statistics - Theory and Methods, 28, 1577-1596. https://doi.org/10.1080/03610929908832374
  12. Kim, C., Kim, W., Park, B.-U. and Lim, J. (2003). Bezier curve smoothing of the Kaplan-Meier estimator, Annals of the Institute of Statistical Mathematics, 55, 359-367.
  13. Kim, C., Kim, S., Park, M. and Lee, H. (2006). A bias reducing technique in kernel distribution function estimation, Computational Statistics, 21, 589-601. https://doi.org/10.1007/s00180-006-0016-x
  14. Kim, C. and Park, J.-H. (2012). On the selection of Bezier points in Bezier curve smoothing, Korean Journal of Applied Statistics, 25, 1049-1058. https://doi.org/10.5351/KJAS.2012.25.6.1049
  15. Loader, C. (1999). Local Regression and Likelihood, Springer, London.
  16. Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators, Scandinavian Journal of Statistics, 9, 65-78.
  17. Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, Chapman and Hall, New York.
  18. Stone, C. J. (1984). An asymptotically optimal window selection rule for kernel density estimates, The Annals of Statistics, 12, 1285-1297. https://doi.org/10.1214/aos/1176346792
  19. Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions, Journal of the Royal Statistical Society, Series B, 36, 111-147.
  20. Wasserman, L. (2006). All of Nonparametric Statistics, Springer, London.

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