Estimation of Densities and Derivatives of Densities with Directional Data

Journal of Multivariate Analysis

Volumn 73, Issue 1, 2000, Pages 18-40

  Klemelä, Jussi ^a  

^a UNIVERSITY OF HELSINKI   (Finland)

Author keywords

Kernel estimator, Laplace operator, Lp error, nonparametric density estimation, plug in method, spherical data

Indexed keywords

EID: 0012933396     PISSN: 0047259X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jmva.1999.1861     Document Type: Article

References (28)

1. Bai Z. D., Radhakrishna Rao C., Zhao L. C. Kernel estimators of density function of directional data. J. Multivariate Anal. 27:1988;24-39.
2. Beran R. Exponential models for directional data. Ann. Statist. 7:1979;1162-1178.
3. Bretagnolle J., Huber C. Estimation des densités: Risque minimax. Z. Wahrsch. Verw. Gebiete. 47:1979;119-137.
4. Cao R., Cuevas A., González-Manteiga W. A comparative study of several smoothing methods in density estimation. Comp. Statist. Data Anal. 17:1994;153-176.
5. Devroye L. A Course in Density Estimation. 1987;Birkhäuser, Boston.
6. 1 View. 1985;Wiley, New York.
7. Diaconis P. Group Representation in Probability and Statistics. IMS Lecture Notes. 11:1988.
8. Engel J., Herrmann E., Gasser T. An iterative band-width selector for kernel estimation of densities and their derivatives. J. Nonparametric Statist. 4:1995;21-34.
9. Fisher N. I., Lewis L., Embleton B. J. J. Statistical Analysis of Spherical Data. 1987;Cambridge Univ. Press.
10. Grund B., Hall P., Marron J. S. Loss and risk in smoothing parameter selection. J. Nonparametric Statist. 4:1994;107-132.
11. Hall P., Sheather S. J., Jones M. C., Marron J. S. On optimal data-based bandwidth selection in kernel density estimation. Biometrika. 78:1991;263-269.
12. 1 distance in nonparametric density estimation. J. Multivariate Anal. 26:1988;59-88.
13. Hall P., Watson G. S., Cabrera J. Kernel density estimation with spherical data. Biometrika. 74:1987;751-762.
14. Hendriks H., Janssen J. H. M., Ruymgaart F. H. Strong uniform convergence of density estimators on compact Euclidian manifolds. Statist. Probab. Lett. 16:1993;305-311.
15. 1 error of a multivariate kernel density estimator. J. Multivariate Anal. 42:1992;245-266.
16. Klemelä J. Rolf Nevanlinna Institute Research Reports A. 1997.
17. Mardia K. V. Statistics of Directional Data. 1972;Academic Press, London.
18. Nadaraya E. A. On the integral mean square error of some nonparametric estimates for the density function. Theor. Probability Appl. 19:1974;133-141.
19. Park B. U., Marron J. S. Comparison of data-driven bandwidth selectors. J. Amer. Statist. Assoc. 85:1990;66-72.
20. Deleted in proof.
21. Ruymgaart F. H. Strong uniform convergence of density estimators on spheres. J. Statist. Plann. Inference. 23:1989;45-52.
22. Scott D. W., Tapia R. A., Thompson J. R. Kernel density estimation revisited. Nonlinear Anal. Theory Meth. Applic. 1:1977;339-372.
23. Sheather S. J., Jones M. C. A reliable data-based band-width selection method for kernel density estimation. J. Roy. Statist. Soc. Ser. B. 53:1991;683-690.
24. Wand M. P., Jones M. C. Multivariate plug-in bandwidth selection. Comp. Statist. 9:1994;97-116.
25. Watson G. S. Orientation statistics in the earth sciences. Bull. Geol. Inst. Univ. Uppsala N.S. 2:1970;73-89.
26. Watson G. S. Statistics on Spheres. University of Arkansas Lecture Notes in Mathematical Sciences. 6:1983;Wiley, New York.
27. Watson G. S., Leadbetter M. R. On the estimation of the probability density, I. Ann. Math. Statist. 34:1963;480-491.
28. Woodroofe M. On choosing a delta-sequence. Ann. Math. Statist. 41:1970;1665-1671.