Bandwidth selections based on cross-validation for estimation of a discontinuity point in density

교차타당성을 이용한 확률밀도함수의 불연속점 추정의 띠폭 선택

  • Huh, Jib (Department of Statistics, Duksung Women's University)
  • 허집 (덕성여자대학교 정보통계학과)
  • Received : 2012.06.27
  • Accepted : 2012.07.19
  • Published : 2012.07.31


The cross-validation is a popular method to select bandwidth in all types of kernel estimation. The maximum likelihood cross-validation, the least squares cross-validation and biased cross-validation have been proposed for bandwidth selection in kernel density estimation. In the case that the probability density function has a discontinuity point, Huh (2012) proposed a method of bandwidth selection using the maximum likelihood cross-validation. In this paper, two forms of cross-validation with the one-sided kernel function are proposed for bandwidth selection to estimate the location and jump size of the discontinuity point of density. These methods are motivated by the least squares cross-validation and the biased cross-validation. By simulated examples, the finite sample performances of two proposed methods with the one of Huh (2012) are compared.

교차타당성은 커널추정량의 평활모수인 띠폭의 선택 방법으로 흔히 활용되고 있다. 연속인 확률밀도함수의 커널추정량의 띠폭 선택으로 널리 쓰이는 교차타당성 방법으로는 최대가능도교차타당성과 더불어 최소제곱교차타당성과 편의교차타당성이 있다. 확률밀도함수가 하나의 불연속점을 가질 때, Huh (2012)는 불연속점 추정을 위한 커널추정량의 띠폭 선택으로 최대가능도교차타당성을 이용한 방법을 제시하였다. 본 연구에서는 Huh (2012)에 의해 최대가능도교차타당성으로 제안된 띠폭선택의 방법과 같이 한쪽방향커널함수를 이용한 최소제곱교차타당성과 편의교차타당성으로 띠폭 선택 방법을 제시하고, 이들 띠폭 선택 방법들과 Huh (2012)의 최대가능도교차타당성을 이용한 띠폭 선택 방법을 모의실험을 통하여 비교연구 하고자 한다.



  1. Cline, D. B. H. and Hart, J. D. (1991). Kernel estimation of densities with discontinuities or discontinuous derivatives. Statistics, 22, 69-84.
  2. Gijbels, I. and Goderniaux, A. C. (2004a). Bandwidth selection for change point estimation in nonparametric regression. Technometrics, 46, 76-86.
  3. Gijbels, I. and Goderniaux, A. C. (2004b). Bootstrap test for change points in nonparametric regression. Journal of Nonparametric Statistics, 16, 591-611.
  4. Gijbels, I. and Goderniaux, A. C. (2005). Data-driven discontinuity detection in derivatives of a regression function. Communications in Statistics-Theory and Methods, 33, 851-871.
  5. Hardle, W. (1991). Smoothing techniques with implementation in S, Springer-Verlag, New York.
  6. Hart, J. D. and Yi, S. (1998). One-sided cross-validation. Journal of the American Statistical Association, 93, 620-631.
  7. Huh, J. (2002). Nonparametric discontinuity point estimation in density or density derivatives. Journal of the Korean Statistical Society, 31, 261-276.
  8. Huh, J. (2007). Nonparametric detection algorithm of discontinuity points in the variance function. Journal of the Korean Data & Information Science Society, 18, 669-678.
  9. Huh, J. (2010a). Estimation of the number of discontinuity points based on likelihood. Journal of the Korean Data & Information Science Society, 21, 51-59.
  10. Huh, J. (2010b). Detection of a change point based on local-likelihood. Journal of Multivariate Analysis, 101, 1681-1700.
  11. Huh, J. (2011). Likelihood based estimation of the log-variance function with a change point. submitted to Journal of Statistical Planning and Inference.
  12. Huh, J. (2012). Bandwidth selection for discontinuity point estimation in density. Journal of the Korean Data & Information Science Society, 23, 79-87
  13. Huh, J. andCarri`ere, K. C. (2002). Estimation of regression functions with a discontinuity in a derivative with local polynomial fits. Statistics and Probability Letters, 56, 329-343.
  14. Huh, J. and Park, B. U. (2004). Detection of change point with local polynomial fits for random design case. Australian and New Zealand Journal of Statistics, 46, 425-441.
  15. Jose, C. T. and Ismail, B. (1999). Change points in nonparametric regression functions. Communication in Statistics-Theory and Methods, 28, 1883-1902.
  16. Kim, J. T., Choi, H. and Huh, J. (2003). Detection of change-points by local linear regression fit. The Korean Communications in Statistics, 10, 31-38.
  17. Loader, C. R. (1996). Change point estimation using nonparametric regression. Annals of Statistics, 24, 1667-1678.
  18. M¨uller, H G. (1992). Change-points in nonparametric regression analysis. Annals of Statistics, 20, 737-761
  19. Otsu, T and Xu, K.-L. (2010). Estimation and inference of discontinuity in density. preprint.
  20. Schuster, E. F. (1985). Incorporating support constraints into nonparametric estimators of densities. Communications in Statistics-Theory and Methods, 14, 1123-1136.
  21. Scott, D. W. and Terrell, G. R. (1987). Biased and unbiased cross-validation in density estimation. Journal of the American Statistical Association, 82, 1131-1146.

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