Asymptotic Analysis of the Jittering Kernel Density Estimator

Abstract

Jittering estimators are nonparametric function estimators for mixed data. They extend arbitrary estimators from the continuous setting by adding random noise to discrete variables. We give an in-depth analysis of the jittering kernel density estimator, which reveals several appealing properties. The estimator is strongly consistent, asymptotically normal, and unbiased for discrete variables. It converges at minimax-optimal rates, which are established as a by-product of our analysis. To understand the effect of adding noise, we further study its asymptotic efficiency and finite sample bias in the univariate discrete case. Simulations show that the estimator is competitive on finite samples. The analysis suggests that similar properties can be expected for other jittering estimators.

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

References

  1. 1.

    I. A. Ahmad and P. B. Cerrito, “Nonparametric Estimation of Joint Discrete-Continuous Probability Densities with Applications”, J. Statist. Planning and Inference 41 (3), 349–364 (1994).

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    S. Efromovich, “Nonparametric Estimation of the Anisotropic Probability Density of Mixed Variables”, J. Multivar. Anal. 102 (3), 468–481 (2011).

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    U. Einmahl and D. M. Mason, “Uniform in Bandwidth Consistency of Kernel-Type Function Estimators,” Ann. Statist. 33 (3), 1380–1403, 06 (2005).

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    C. Genest, J. G. Nešlehová, and B. Rémillard, “Asymptotic Behavior of the Empirical Multilinear Copula Process under Broad Conditions”, J. Multivar. Anal. (2017).

    Google Scholar 

  5. 5.

    P. Hall, J. Racine, and Q. Li, “Cross-Validation and the Estimation of Conditional Probability Densities”, J. Amer. Statist. Assoc. 99 (468), 1015–1026 (2004).

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    P. Hall et al., “Orthogonal SeriesMethods for Both Qualitative and Quantitative Data”, Ann. Statist. 11 (3), 1004–1007 (1983).

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Y. Han, J. Jiao, and T. Weissman, “Minimax Estimation of Discrete Distributions under Loss”, IEEE Trans. on Information Theory 61 (11), 6343–6354 (2015).

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    T. Hayfield and J. S. Racine, “Nonparametric Econometrics: The np Package”, J. Statist. Software 27 (5), (2008).

    Google Scholar 

  9. 9.

    I. Ibragimov and R. Khas’minskii, “Estimation of Distribution Density”, J. Soviet Math. 21 (1), 40–57 (1983).

    Article  MATH  Google Scholar 

  10. 10.

    C. G. G. Aitken and J. Aitchison, “Multivariate Binary Discrimination by the Kernel Method”, Biometrika 63 (3), 413–420 (1976).

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Q. Li and J. Racine, “Nonparametric Estimation of Distributions with Categorical and Continuous Data”, J. Multivar. Anal. 86 (2), 266–292 (2003).

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    J. S. Marron, “Visual Understanding of Higher-Order Kernels”, J. Comput. and Graph. Statist. 3 (4), 447–458 (1994).

    Google Scholar 

  13. 13.

    J. F. Monahan, Numerical Methods of Statistics. (Cambridge University Press, Cambridge, 2011).

    Google Scholar 

  14. 14.

    T. Nagler, “A Generic Approach to Nonparametric Function Estimation with Mixed Data”, Statist. Probab. Lett., DOI 10.1016/j.spl.2018.02.040 (in press).

  15. 15.

    E. Parzen, “On Estimation of a Probability Density Function and Mode”, Ann.Math. Statist. 33 (3), 1065–1076, 09 (1962).

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    J. Racine and Q. Li, “Nonparametric Estimation of Regression Functions with Both Categorical and Continuous Data”, J. Econometrics 119 (1), 99–130 (2004).

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    M. Rosenblatt, “Remarks on Some Nonparametric Estimates of a Density Function”, Ann. Math. Statist. 27 (3), 832–837, 09 (1956).

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    D. W. Scott, “The Curse of Dimensionality and Dimension Reduction”, in Multivariate Density Estimation: Theory, Practice, and Visualization (Wiley, New York, 2008) pp. 195–217.

    Google Scholar 

  19. 19.

    L. Simar, V. Zelenyuk, et al., To Smooth or not to Smooth? The Case of Discrete Variables in Nonparametric Regressions, in Working Paper Series (Center for Efficiency and Productivity Analysis, 2011).

    Google Scholar 

  20. 20.

    C. J. Stone, “Optimal Rates of Convergence for Nonparametric Estimators”, Ann. Statist. 8 (6), 1348–1360, 11 (1980).

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    C. J. Stone, “Optimal Uniform Rate of Convergence for Nonparametric Estimators of a Density Function or Its Derivatives”, Recent Advances in Statist. 5 (1983).

  22. 22.

    A. Tsybakov, Introduction to Nonparametric Estimation, in Springer Series in Statist. (Springer, New York, 2008).

    Google Scholar 

  23. 23.

    M. Wand, “Error Analysis for General Multivariate Kernel Estimators”, J. Nonparam. Statist. 2 (1), 1–15 (1992).

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to T. Nagler.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nagler, T. Asymptotic Analysis of the Jittering Kernel Density Estimator. Math. Meth. Stat. 27, 32–46 (2018). https://doi.org/10.3103/S1066530718010027

Download citation

Keywords

  • density
  • discrete
  • jittering
  • kernel
  • minimax
  • mixed data

2000 Mathematics Subject Classification

  • 62G07