Skip to main content

Probability Density Function Estimation Using Gamma Kernels

Abstract

We consider estimating density functions which have support on [0, ∞) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n −4/5 x −1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.

This is a preview of subscription content, access via your institution.

References

  1. Brown, B. M. and Chen, S. X. (1999). Beta-Bernstein smoothing for regression curves with compact supports, Scand. J. Statist., 26, 47–59.

    Google Scholar 

  2. Chen, S. X. (1999). Beta kernel estimators for density functions, Comput. Statist. Data Anal., 31, 131–145.

    Google Scholar 

  3. Chen, S. X. (2000). Beta kernel smoothers for regression curves, Statist. Sinica, 10, 73–91.

    Google Scholar 

  4. Cheng, M. Y., Fan, J. and Marron, J. S. (1997). On automatic boundary corrections, Ann. Statist., 25, 1691–1708.

    Google Scholar 

  5. Cowling, A. and Hall, P. (1996). On pseudodata methods for removing boundary effects in kernel density estimation, J. R. Statist. Soc. B, 58, 551–563.

    Google Scholar 

  6. Hall, P. and Wehrly, T. E. (1991). A geometrical method for removing edge effects from kernel-type nonparametric regression estimators, J. Amer. Statist. Assoc., 86, 665–672.

    Google Scholar 

  7. Jones, M. C. (1993). Simple boundary correction for kernel density estimation, Statist. Comput., 3, 135–146.

    Google Scholar 

  8. Jones, M. C. and Foster, P. J. (1996). A simple nonnegative boundary correction method for kernel density estimation, Statist. Sinica., 6, 1005–1013.

    Google Scholar 

  9. Lejeune, M. and Sarda, P. (1992). Smooth estimators of distribution and density functions, Comput. Statist. Data Anal., 14, 457–471.

    Google Scholar 

  10. Marron, J. S. and Ruppert, D. (1994). Transformation to reduce boundary bias in kernel density estimation, J. R. Statist. Soc. B, 56, 653–671.

    Google Scholar 

  11. Müller, H.-G. (1991). Smooth optimum kernel estimators near endpoints, Biometrika, 78, 521–530.

    Google Scholar 

  12. Müller, H.-G. (1993). On the boundary kernel method for non-parametric curve estimation near end-points. Scand. J. Statist., 20, 313–328.

    Google Scholar 

  13. Müller, H.-G. and Wang, J.-L. (1994). Hazard rate estimation under random censoring with varying kernels and bandwidths. Biometrics, 50, 61–76.

    Google Scholar 

  14. Müller, H.-G. and Zhou, H. (1991). A discussion to “Transformations in density estimation” by Wand, M., Marron, J. S. and Ruppert, D., J. Amer. Statist. Assoc., 86, 356–358.

    Google Scholar 

  15. Press, W. H., Flannery, F., Teukolsky, S. A. and Vettering, W. T. (1992). Numerical Recipes: the Art of Scientific Computing, Cambridge University Press, Cambridge.

    Google Scholar 

  16. Schuster, E. F. (1985). Incorporating support constraints into nonparametric estimators of densities, Commun. Statist. Theory Methods, 14, 1123–1136.

    Google Scholar 

  17. Silverman, B. W. (1986). Density Estimation, Chapman and Hall, London.

    Google Scholar 

  18. Wand, M. and Jones, M. C. (1995). Kernel Smoothing, Chapman and Hall, London.

    Google Scholar 

Download references

Author information

Affiliations

Authors

About this article

Cite this article

Chen, S.X. Probability Density Function Estimation Using Gamma Kernels. Annals of the Institute of Statistical Mathematics 52, 471–480 (2000). https://doi.org/10.1023/A:1004165218295

Download citation

  • Boundary bias
  • gamma kernels
  • local linear estimators
  • variable kernels