Multivariate Density Estimation with General Flat-Top Kernels of Infinite Order

Journal of Multivariate Analysis

Volumn 68, Issue 1, 1999, Pages 1-25

  Politis, Dimitris N ^a   Romano, Joseph P ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b NONE

Author keywords

Bias reduction; Fourier transform; Kernel; Mean squared error; Nonparametric density estimation; Rate of convergence; Smoothing

Indexed keywords

EID: 0042995281     PISSN: 0047259X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmva.1998.1774     Document Type: Article

References (30)

1. Barlett M. S. Statistical estimation of density functions. Sankhya, Ser. A. 25:1963;245-254.
2. Butzer P., Nessel R. Fourier Analysis and Approximation. 1971;Academic Press, New York.
3. Cacoullos T. Estimation of a multivariate density. Ann. Inst. Statist. Math. 18:1966;178-189.
4. Davis K. B. Mean integrated square error properties of density estimates. Ann. Statist. 5:1977;530-535.
5. Devroye L. A Course in Density Estimation. 1987;Birkhäuser, Boston.
6. Devroye L. A note on the usefulness of superkernels in density estimates. Ann. Statist. 20:1992;2037-2056.
7. Gajek L. On improving density estimators which are not bona fide functions. Ann. Statist. 14:1968;1612-1618.
8. Gasser T., Müller H. G., Mammitzsch V. Kernels for nonparametric curve estimation. J. Roy. Statist. Soc. B. 47:1985;238-252.
9. Granovsky B. L., Müller H. G. Optimal kernel methods: A unifying variational principle. Internat. Statist. Rev. 59:1991;373-388.
10. Györfi L., Härdle W., Sarda P., Vieu P. Nonparametric Curve Estimation from Time Series. Lecture Notes in Statistics No. 60. 1989;Springer-Verlag, Berlin/New York.
11. Hall P., Murison R. D. Correcting the negativity of high-order kernel density estimators. J. Multivar. Anal. 47:1992;103-122.
12. Ibragimov I. A., Hasminksii R. Z. Estimation of distribution density belonging to a class of entire functions. Theor. Probab. Appl. 27:1982;551-562.
13. Jones M. C. On higher order kernels. J. Nonparametr. Statist. 5:1995;215-221.
14. Jones M. C., Foster P. J. Generalized jacknifing and higher order kernels. J. Nonparametr. Statist. 3:1993;81-94.
15. Katznelson Y. An Introduction to Harmonic Analysis. 1968;Dover, New York.
16. Marron J. S. Visual understanding of higher order kernels. J. Comput. Graphical Statist. 3:1994;447-458.
17. Marron J. S., Wand M. P. Exact mean integrated squared error. Ann. Statist. 20:1992;712-736.
18. Müller H. G. Nonparametric Regression Analysis of Longitudinal Data. 1988;Springer-Verlag, Berlin.
19. Nadaraya E. A. Nonparametric Estimation of Probability Densities and Regression Curves. 1989;Kluwer Academic, Dordrecht.
20. Parzen E. On estimation of a probability density function and its mode. Ann. Math. Statist. 33:1962;1065-1076.
21. Politis D. N., Romano J. P. On a family of smoothing kernels of infinite order. Computing Science and Statistics, Proceedings of the 25th Symposium on the interface. 1993;The Interface Foundation of North America, San Diego.
22. Politis D. N., Romano J. P. On flat-top kernel spectral density estimators for homogeneous random fields. J. Statist. Plann. Inference. 51:1996;41-53.
23. Priestley M. B. Spectral Analysis and Time Series. 1981;Academic Press, New York.
24. Rosenblatt M. Stochastic Curve Estimation. NSF-CBMS Regional Conference Series. 3:1991;Institute of Mathematical Statistics, Hayward.
25. Scott D. W. Multivariate Density Estimation: Theory, Practice, and Visualization. 1992;Wiley, New York.
26. Silverman B. W. Density Estimation for Statistics and Data Analysis. 1986;Chapman and Hall, London.
27. Stein E. M., Weiss W. Introduction to Fourier Analysis on Euclidean Spaces. 1971;Princeton Univ. Press, Princeton.
28. Wahba G. Optimal convergence properties of variable knot, kernel and orthogonal series methods for density estimation. Ann. Statist. 3:1975;15-29.
29. Wand M. P., Jones M. C. Comparison of smoothing parameterizations in bivariate kernel density estimation. J. Amer. Statist. Assoc. 88:1993;520-528.
30. Watson G. S., Leadbetter M. R. On the estimation of the probability density, I. Ann. Math. Statist. 33:1963;480-491.