Kernel density estimation on Riemannian manifolds

Statistics and Probability Letters

Volumn 73, Issue 3, 2005, Pages 297-304

  Pelletier, Bruno ^a  

^a UNIVERSITÉ DU HAVRE   (France)

Author keywords

Kernel density estimation; L2 convergence; Nonparametric density estimation; Riemannian manifolds

Indexed keywords

EID: 20744434706     PISSN: 01677152     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spl.2005.04.004     Document Type: Article

References (31)

1. H. Akaike An approximation to the density function Ann. Inst. Statist. Math. 6 1954 127-132
2. A. Besse Manifolds all of those geodesics are closed. Ergebnisse der Mathematik und ihrer Grenzgebiete 93 1978 Springer Berlin
3. R. Bhattacharya V. Patrangenaru Nonparametric estimation of location and dispersion on Riemannian manifolds J. Statist. Plann. Inference 108 2002 23-35
4. R. Bhattacharya V. Patrangenaru Large sample theory of intrinsic and extrinsic sample means on manifold (part i) Ann. Statist. 31 1 2003 1-29
5. W. Boothby An Introduction to Differentiable Manifolds and Riemannian Geometry 1975 Academic Press New York
6. Chavel, I., 1993. Riemannian geometry: A modern introduction. In: Cambridge Tracts in Mathematics, vol. 108. Cambridge University Press, Cambridge.
7. J. Corcuera W. Kendall Riemannian barycentres and geodesic convexity Math. Proc. Cambridge Philos. Soc. 127 2 1999 253-269
8. 1 -View 1985 Wiley New York
9. Emery, M., Mokobodzki, G., 1991. Sur le barycentre d'une probabilité dans une variété. In: Séminaire de Probabilité, XXV. Lecture Notes in Mathematics, vol. 1485. Springer, Berlin, pp. 220-233.
10. N. Fischer T. Lewis B. Embleton Statistical Analysis of Spherical Data 1993 Cambridge University Press Cambridge
11. P. Hall G. Watson J. Cabrera Kernel density estimation with spherical data Biometrika 74 1987 751-762
12. D. Healy P. Kim An empirical Bayes approach to directional data and efficient computation on the sphere Ann. Statist. 24 1996 232-254
13. D. Healy H. Hendriks P. Kim Spherical deconvolution J. Multivariate Anal. 67 1998 1-22
14. E. Hebey Introduction à l'analyse non linéaire sur les variétés 1997 Diderot Paris, New York
15. H. Hendriks Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions Ann. Statist. 18 1990 832-849
16. H. Hendriks Application of fast spherical Fourier transform to density estimation J. Multivariate Anal. 84 2003 209-221
17. H. Hendriks J. Janssen F. Ruymgaart Strong uniform convergence of density estimators on compact Euclidean manifolds Statist. Probab. Lett. 16 1993 305-311
18. P. Jupp K. Mardia A unified view of the theory of directional statistics 1975-1988 Internat. Statist. Rev. 57 1989 261-294
19. H. Karcher Riemannian center of mass and mollifier smoothing Comm. Pure Appl. Math. 30 1977 509-554
20. Kobayashi, S., Nomizu, K., 1969. Foundations of Differential Geometry, vols. 1,2. Wiley, New York.
21. H. Le On the consistency of procrustean mean shapes Adv. Appl. Probab. 30 1998 53-63
22. J. Lee F. Ruymgaart Nonparametric curve estimation on Stiefel manifolds Nonparam. Statist. 6 1996 57-68
23. Lee, J., Paige, R., Patrangenaru, V., Ruymgaart, F., 2004. Nonparametric density estimation on homogeneous spaces in high level image analysis. In: Aykroyd, R., Barber, S., Mardia, K. (Eds.), Bioinformatics, Images, and Wavelets. pp. 37-40.
24. K. Mardia Statistics of Directional Data 1972 Academic Press New York
25. J. Oller J. Corcuera Intrinsic analysis of statistical estimation Ann. Statist. 23 5 1995 1562-1581
26. E. Parzen On the estimation of a probability density function and the mode Ann. Math. Statist. 33 1962 1065-1076
27. X. Pennec N. Ayache Uniform distribution, distance and expectation problems for geometric features processing J. Math. Imaging Vision 9 1 1998 49-67
28. M. Rosenblatt Remarks on some nonparametric estimates of a density function Ann. Math. Statist. 27 1956 832-837
29. A. Van der Vaart Asymptotic Statistics 1998 Cambridge University Press Cambridge
30. G. Watson Statistics on Spheres 1983 Wiley New York
31. T. Willmore Riemannian Geometry 1993 Oxford University Press Oxford