A class of unbiased kernel estimates of a probability density function

• Autorzy: Tomasz Rychlik
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

rectangular kernel; kernel function; randomized estimate; probability density function; nonparametric estimate; unbiased estimate

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 1993-1995
• Tom: 22
• Numer: 4
• Strony: 485-497

Daty

wydano

1995

otrzymano

1994-01-10

Twórcy

autor

Tomasz Rychlik

• Institute of Mathematics, Polish Academy of Sciences, Abrahama 18, 81-825 Sopot, Poland

Bibliografia

• [1] M. S. Bartlett, Statistical estimation of density funtions, Sankhyā Ser. A 25 (1963), 245-254.
• [2] P. Bickel and E. Lehmann, Unbiased estimation in convex families, Ann. Math. Statist. 40 (1969), 1523-1535.
• [3] N. N. Chentsov, An estimate of an unknown probability density under observations, Dokl. Akad. Nauk SSSR 147 (1962), 45-48 (in Russian).
• [4] L. P. Devroye, A Course in Density Estimation, Birkhäuser, Boston, 1987.
• [5] L. P. Devroye and L. Győrfi, Nonparametric Density Estimation. The L_1 View, Wiley, New York, 1985.
• [6] L. P. Devroye and T. J. Wagner, The L_1 convergence of kernel density estimates, Ann. Statist. 7 (1979), 1136-1139.
• [7] H. Doss and J. Sethuraman, The price of bias reduction when there is no unbiased estimate, ibid. 17 (1989), 440-442.
• [8] L. Gajek, On improving density estimators which are not bona fide functions, ibid. 14 (1986), 1612-1618.
• [9] J. Koronacki, Kernel estimation of smooth densities using Fabian's approach, Statistics 18 (1987), 37-47.
• [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.
• [11] R. C. Liu and L. D. Brown, Nonexistence of informative unbiased estimators in singular problems, Ann. Statist. 21 (1993), 1-13.
• [12] E. Parzen, On estimation of a probability density function and mode, Ann. Math. Statist. 33 (1962), 1065-1076.
• [13] M. Rosenblatt, Remarks on some nonparametric estimates of a density function, ibid. 27 (1956), 832-837.
• [14] T. Rychlik, Unbiased nonparametric estimation of the derivative of the mean, Statist. Probab. Lett. 10 (1990), 329-333.
• [15] W. R. Schucany and J. P. Sommers, Improvement of kernel type density estimators, J. Amer. Statist. Assoc. 72 (1977), 420-423.
• [16] E. F. Schuster, Estimation of a probability density function and its derivatives, Ann. Math. Statist. 40 (1969), 1187-1195.
• [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.
• [18] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970.
• [19] V. G. Voinov and M. S. Nikulin, Unbiased Estimators and their Applications, Vol. 1, Univariate Case, Kluwer Academic Publ., Dordrecht, 1993.
• [20] H. Yamato, Some statistical properties of estimators of density and distribution functions, Bull. Math. Statist. 15 (1972), 113-131.