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On the convergence of kernel estimators of probability density functions

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Summary

The properties of the characteristic function of the fixed-bandwidth kernel estimator of a probability density function are used to derive estimates of the rate of almost sure convergence of such estimators in a family of Hilbert spaces. The convergence of these estimators in a reproducing-kernel Hilbert space is used to prove the uniform convergence of variable-bandwidth estimators. An algorithm employing the fast Fourier transform and heuristic estimates of the optimal bandwidth is presented, and numerical experiments using four density functions are described.

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References

  1. Brieman, L., Meisel, W., and Purcell, E. (1977). Variable kernel estimates of multivariate densities,Technometrics,19, 135–144.

    Article  Google Scholar 

  2. Davis, K. B. (1975). Mean square error properties of density estimates.Ann. Statist.,3, 1025–1030.

    Article  MathSciNet  Google Scholar 

  3. Epanechnikov, V. A. (1969). Nonparametric estimators of a multivariate probability density,Theory Prob. Appl.,14, 153–158.

    Article  MathSciNet  Google Scholar 

  4. Farrell, R. H. (1972). On the best obtainable asymptotic rates of convergence in the estimation of a density function at a point,Ann. Math. Statist.,43, 170–180.

    Article  MathSciNet  Google Scholar 

  5. Loftsgaarden, D. O. and Quensenberry, C. P. (1965). A non-parametric estimate of a multivariate density function,Ann. Math. Statist.,36, 1049–1051.

    Article  MathSciNet  Google Scholar 

  6. Marcinkiewicz, J. and Zygmund, A. (1937). Sur les fonctions indépendantes,Fundamenta Math.,29, 60–90.

    MATH  Google Scholar 

  7. Moore, D. S. and Yackel, J. W. (1977). Consistency properties of nearest neighbor density function estimators,Ann. Statist.,5, 143–154.

    Article  MathSciNet  Google Scholar 

  8. Nadaraya, E. A. (1965) On nonparametric estimates of density functions and regression curves.Theory Prob. Appl.,9, 141–142.

    Article  Google Scholar 

  9. Nadaraya, E. A. (1973). On convergence inL 2-norm of probability estimates,Theory Prob. Appl.,18, 808–811.

    Article  Google Scholar 

  10. Parzen, E. (1962). On the estimation of a probability density function and the mode,Ann. Math. Statist.,33, 1065–1076.

    Article  MathSciNet  Google Scholar 

  11. Scott, D. W. (1976). Nonparametric density estimation by optimization theoretic techniques, Doctoral dissertation at Rice University, Houston, Texas.

  12. Tapia, R. A. and Thompson, J. R. (1978).Nonparametric Probability Density Estimation, Johns Hopkins University Press, Baltimore, Maryland.

    MATH  Google Scholar 

  13. Taylor, R. L. Complete convergence for weighted sums of arrays of random elements, preprint.

  14. Taylor, R. L. Convergence of weighted sums of arrays of random elements in typep spaces with application to density estimation, preprint.

  15. Woyczynski, W. A. On Marcinkiewicz-Zygmund laws of large numbers in Banach spaces and related rates of convergence, preprint.

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Authors and Affiliations

  1. Florida Mental Health Institute, University of South Florida, South Florida, USA

    Albert E. Rust & Chris P. Tsokos

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  1. Albert E. Rust
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  2. Chris P. Tsokos
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Additional information

This research was supported by the United States Air Force, Air Force Office of Scientific Research, Under Grant Number AFOSR-76-2711.

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Rust, A.E., Tsokos, C.P. On the convergence of kernel estimators of probability density functions. Ann Inst Stat Math 33, 233–246 (1981). https://doi.org/10.1007/BF02480937

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  • DOI: https://doi.org/10.1007/BF02480937

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