A class of unbiased kernel estimates of a probability density function

Tomasz Rychlik

Applicationes Mathematicae (1995)

  • Volume: 22, Issue: 4, page 485-497
  • ISSN: 1233-7234

Abstract

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We propose a class of unbiased and strongly consistent nonparametric kernel estimates of a probability density function, based on a random choice of the sample size and the kernel function. The expected sample size can be arbitrarily small and mild conditions on the local behavior of the density function are imposed.

How to cite

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Rychlik, Tomasz. "A class of unbiased kernel estimates of a probability density function." Applicationes Mathematicae 22.4 (1995): 485-497. <http://eudml.org/doc/219109>.

@article{Rychlik1995,
abstract = {We propose a class of unbiased and strongly consistent nonparametric kernel estimates of a probability density function, based on a random choice of the sample size and the kernel function. The expected sample size can be arbitrarily small and mild conditions on the local behavior of the density function are imposed.},
author = {Rychlik, Tomasz},
journal = {Applicationes Mathematicae},
keywords = {rectangular kernel; kernel function; randomized estimate; probability density function; nonparametric estimate; unbiased estimate; density estimation; unbiasedness; strong consistency; kernel estimates; local behavior},
language = {eng},
number = {4},
pages = {485-497},
title = {A class of unbiased kernel estimates of a probability density function},
url = {http://eudml.org/doc/219109},
volume = {22},
year = {1995},
}

TY - JOUR
AU - Rychlik, Tomasz
TI - A class of unbiased kernel estimates of a probability density function
JO - Applicationes Mathematicae
PY - 1995
VL - 22
IS - 4
SP - 485
EP - 497
AB - We propose a class of unbiased and strongly consistent nonparametric kernel estimates of a probability density function, based on a random choice of the sample size and the kernel function. The expected sample size can be arbitrarily small and mild conditions on the local behavior of the density function are imposed.
LA - eng
KW - rectangular kernel; kernel function; randomized estimate; probability density function; nonparametric estimate; unbiased estimate; density estimation; unbiasedness; strong consistency; kernel estimates; local behavior
UR - http://eudml.org/doc/219109
ER -

References

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  1. [1] M. S. Bartlett, Statistical estimation of density funtions, Sankhyā Ser. A 25 (1963), 245-254. Zbl0129.32302
  2. [2] P. Bickel and E. Lehmann, Unbiased estimation in convex families, Ann. Math. Statist. 40 (1969), 1523-1535. Zbl0197.44602
  3. [3] N. N. Chentsov, An estimate of an unknown probability density under observations, Dokl. Akad. Nauk SSSR 147 (1962), 45-48 (in Russian). 
  4. [4] L. P. Devroye, A Course in Density Estimation, Birkhäuser, Boston, 1987. Zbl0617.62043
  5. [5] L. P. Devroye and L. Győrfi, Nonparametric Density Estimation. The L_1 View, Wiley, New York, 1985. Zbl0546.62015
  6. [6] L. P. Devroye and T. J. Wagner, The L_1 convergence of kernel density estimates, Ann. Statist. 7 (1979), 1136-1139. Zbl0423.62031
  7. [7] H. Doss and J. Sethuraman, The price of bias reduction when there is no unbiased estimate, ibid. 17 (1989), 440-442. Zbl0669.62010
  8. [8] L. Gajek, On improving density estimators which are not bona fide functions, ibid. 14 (1986), 1612-1618. Zbl0623.62034
  9. [9] J. Koronacki, Kernel estimation of smooth densities using Fabian's approach, Statistics 18 (1987), 37-47. Zbl0612.62054
  10. [10] R. Kronmal and M. Tarter, The estimation of probability densities and cumulatives by Fourier series methods, J. Amer. Statist. Assoc. 63 (1968), 925-952. Zbl0169.21403
  11. [11] R. C. Liu and L. D. Brown, Nonexistence of informative unbiased estimators in singular problems, Ann. Statist. 21 (1993), 1-13. Zbl0783.62026
  12. [12] E. Parzen, On estimation of a probability density function and mode, Ann. Math. Statist. 33 (1962), 1065-1076. Zbl0116.11302
  13. [13] M. Rosenblatt, Remarks on some nonparametric estimates of a density function, ibid. 27 (1956), 832-837. Zbl0073.14602
  14. [14] T. Rychlik, Unbiased nonparametric estimation of the derivative of the mean, Statist. Probab. Lett. 10 (1990), 329-333. Zbl0703.62049
  15. [15] W. R. Schucany and J. P. Sommers, Improvement of kernel type density estimators, J. Amer. Statist. Assoc. 72 (1977), 420-423. Zbl0369.62039
  16. [16] E. F. Schuster, Estimation of a probability density function and its derivatives, Ann. Math. Statist. 40 (1969), 1187-1195. Zbl0212.21703
  17. [17] B. W. Silverman, Weak and strong uniform consistency of the kernel estimate of a density and its derivatives, Ann. Statist. 6 (1978), 177-184. Zbl0376.62024
  18. [18] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. Zbl0207.13501
  19. [19] V. G. Voinov and M. S. Nikulin, Unbiased Estimators and their Applications, Vol. 1, Univariate Case, Kluwer Academic Publ., Dordrecht, 1993. Zbl0832.62019
  20. [20] H. Yamato, Some statistical properties of estimators of density and distribution functions, Bull. Math. Statist. 15 (1972), 113-131. Zbl0259.62036

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