Research Article
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Year 2019, Volume: 48 Issue: 2, 616 - 625, 01.04.2019

Abstract

References

  • Abramson, I.S. On bandwidth variation in kernel estimates-a square root law, Annals of Statistics 10, 1217–1223, 1982.
  • Bowman, A. An alternative method of cross-validation for the smoothing of density estimates, Biometrika 71, 353–360, 1984.
  • Breiman, L., Meisel, W., and Purcell, E. Variable kernel estimates of multivariate densities, Technometrics 19, 135–144, 1977.
  • Chiu, T. A comparative review of bandwidth selection for kernel density estimation, Statistica Sinica 6, 129–145, 1996.
  • Core Team, R. R: a language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, 2014.
  • Cula, S., Demir, S., and Toktamis, O. The Finite Sample Performance of Modified Adaptive Kernel Estimators for Probability Density Function, J Sci Res Rep, 11 (5), 1–9, 2016.
  • Hall, P. and Marron, J.S. Variable window width kernel estimates of probability densities, Probability Theory and Related Fields, 80, 37–49, 1988.
  • Hall, P. and Marron, J.S. Local minima in cross-validation functions, Journal of the Royal Statistical Society, Series B, 53, 245–252, 1991.
  • Hardle, W. Smoothing Techniques with Implementation in S (Springer-Verlag, 1991).
  • Jones, M.C. Variable Kernel Density Estimation and Variable Kernel Density Estimation, Australian Journal of Statistics, 32 (3), 361–371, 1990.
  • Loader, C.R. Bandwidth selection: classical or plug-in, Annals of Statistics 27, 415–438, 1999.
  • Park, B. U. and Marron J. S. Comparison of data-driven bandwidth selectors. Journal of the American Statistical Association, 85, 66–72, 1990.
  • Scott, D. W. Multivariate Density Estimation (Wiley, 1992).
  • Scott, D.W. and Terrell, G.R. Biased and unbiased cross-validation in density estimation, Journal of the American Statistical Association 82 (400), 1131–1146, 1987.
  • Sheather, S.J. and Jones M.C. A reliable data-based bandwidth selection method for kernel density estimation, Journal of the Royal Statistical Society Series B, 53, 683–690, 1991.
  • Sheather, S.J. Density estimation, Statistical Science 19 (4), 588–597, 2004.
  • Silverman, B. W. Density Estimation for Statistics and Data Analysis (Chapman&Hall, 1996).
  • Terrell, G.R. and Scott, D.W. Variable kernel density estimation, Annals of Statistics 20 (3), 1236–1265, 1992.
  • Van Kerm, P. Adaptive kernel density estimation, Stata Journal, 3, 148–156, 2003.
  • Wand, M. P. and Jones, M. C. Kernel Smoothing (Chapman&Hall, 1995).
  • Zhang, J Adaptive normal reference bandwidth based on quantile for kernel density estimation, Journal of Applied Statistics, 38 (12), 2869–2880, 2011.
  • Zhang, J Generalized least squares cross-validation in kernel density estimation, Statistica Neerlandica, 69, 315–328, 2015.

Adaptive kernel density estimation with generalized least square cross-validation

Year 2019, Volume: 48 Issue: 2, 616 - 625, 01.04.2019

Abstract

Adaptive kernel density estimator is an efficient estimator when the density to be estimated has long tail or multi-mode. They use varying bandwidths at each observation point by adapting a fixed bandwidth for data. It is well-known that bandwidth selection is too important for performance of kernel estimators. An efficient recent method is the generalized least square cross-validation which improves the least squares cross-validation. In this paper, performances of the adaptive kernel estimators obtained based on the generalized least square cross-validation are investigated. We performed a simulation study to inform about performances of the modified adaptive kernel estimators. For the simulation, we use also the bandwidth selection methods of normal reference, least squares cross-validation, biased cross-validation, and plug-in methods. Simulation study shows that the adaptive kernel estimators improve the performances of the kernel estimators with fixed bandwidth selected based on generalized least square cross-validation.

References

  • Abramson, I.S. On bandwidth variation in kernel estimates-a square root law, Annals of Statistics 10, 1217–1223, 1982.
  • Bowman, A. An alternative method of cross-validation for the smoothing of density estimates, Biometrika 71, 353–360, 1984.
  • Breiman, L., Meisel, W., and Purcell, E. Variable kernel estimates of multivariate densities, Technometrics 19, 135–144, 1977.
  • Chiu, T. A comparative review of bandwidth selection for kernel density estimation, Statistica Sinica 6, 129–145, 1996.
  • Core Team, R. R: a language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, 2014.
  • Cula, S., Demir, S., and Toktamis, O. The Finite Sample Performance of Modified Adaptive Kernel Estimators for Probability Density Function, J Sci Res Rep, 11 (5), 1–9, 2016.
  • Hall, P. and Marron, J.S. Variable window width kernel estimates of probability densities, Probability Theory and Related Fields, 80, 37–49, 1988.
  • Hall, P. and Marron, J.S. Local minima in cross-validation functions, Journal of the Royal Statistical Society, Series B, 53, 245–252, 1991.
  • Hardle, W. Smoothing Techniques with Implementation in S (Springer-Verlag, 1991).
  • Jones, M.C. Variable Kernel Density Estimation and Variable Kernel Density Estimation, Australian Journal of Statistics, 32 (3), 361–371, 1990.
  • Loader, C.R. Bandwidth selection: classical or plug-in, Annals of Statistics 27, 415–438, 1999.
  • Park, B. U. and Marron J. S. Comparison of data-driven bandwidth selectors. Journal of the American Statistical Association, 85, 66–72, 1990.
  • Scott, D. W. Multivariate Density Estimation (Wiley, 1992).
  • Scott, D.W. and Terrell, G.R. Biased and unbiased cross-validation in density estimation, Journal of the American Statistical Association 82 (400), 1131–1146, 1987.
  • Sheather, S.J. and Jones M.C. A reliable data-based bandwidth selection method for kernel density estimation, Journal of the Royal Statistical Society Series B, 53, 683–690, 1991.
  • Sheather, S.J. Density estimation, Statistical Science 19 (4), 588–597, 2004.
  • Silverman, B. W. Density Estimation for Statistics and Data Analysis (Chapman&Hall, 1996).
  • Terrell, G.R. and Scott, D.W. Variable kernel density estimation, Annals of Statistics 20 (3), 1236–1265, 1992.
  • Van Kerm, P. Adaptive kernel density estimation, Stata Journal, 3, 148–156, 2003.
  • Wand, M. P. and Jones, M. C. Kernel Smoothing (Chapman&Hall, 1995).
  • Zhang, J Adaptive normal reference bandwidth based on quantile for kernel density estimation, Journal of Applied Statistics, 38 (12), 2869–2880, 2011.
  • Zhang, J Generalized least squares cross-validation in kernel density estimation, Statistica Neerlandica, 69, 315–328, 2015.
There are 22 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Statistics
Authors

Serdar Demir 0000-0002-7504-6383

Publication Date April 1, 2019
Published in Issue Year 2019 Volume: 48 Issue: 2

Cite

APA Demir, S. (2019). Adaptive kernel density estimation with generalized least square cross-validation. Hacettepe Journal of Mathematics and Statistics, 48(2), 616-625.
AMA Demir S. Adaptive kernel density estimation with generalized least square cross-validation. Hacettepe Journal of Mathematics and Statistics. April 2019;48(2):616-625.
Chicago Demir, Serdar. “Adaptive Kernel Density Estimation With Generalized Least Square Cross-Validation”. Hacettepe Journal of Mathematics and Statistics 48, no. 2 (April 2019): 616-25.
EndNote Demir S (April 1, 2019) Adaptive kernel density estimation with generalized least square cross-validation. Hacettepe Journal of Mathematics and Statistics 48 2 616–625.
IEEE S. Demir, “Adaptive kernel density estimation with generalized least square cross-validation”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 2, pp. 616–625, 2019.
ISNAD Demir, Serdar. “Adaptive Kernel Density Estimation With Generalized Least Square Cross-Validation”. Hacettepe Journal of Mathematics and Statistics 48/2 (April 2019), 616-625.
JAMA Demir S. Adaptive kernel density estimation with generalized least square cross-validation. Hacettepe Journal of Mathematics and Statistics. 2019;48:616–625.
MLA Demir, Serdar. “Adaptive Kernel Density Estimation With Generalized Least Square Cross-Validation”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 2, 2019, pp. 616-25.
Vancouver Demir S. Adaptive kernel density estimation with generalized least square cross-validation. Hacettepe Journal of Mathematics and Statistics. 2019;48(2):616-25.