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Kernel density estimation on Riemannian manifolds. (English) Zbl 1065.62063

Summary: The estimation of the underlying probability density of \(n\) i.i.d. random objects on a compact Riemannian manifold without boundary is considered. The proposed methodology adapts the technique of kernel density estimation on Euclidean sample spaces to this non-Euclidean setting. Under sufficient regularity assumptions on the underlying density, \(L^2\) convergence rates are obtained.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
53A99 Classical differential geometry
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[1] Akaike, H., An approximation to the density function, Ann. inst. statist. math., 6, 127-132, (1954) · Zbl 0058.12302
[2] Besse, A., Manifolds all of those geodesics are closed. ergebnisse der Mathematik und ihrer grenzgebiete 93, (1978), Springer Berlin
[3] Bhattacharya, R.; Patrangenaru, V., Nonparametric estimation of location and dispersion on Riemannian manifolds, J. statist. plann. inference, 108, 23-35, (2002) · Zbl 1031.62024
[4] Bhattacharya, R.; Patrangenaru, V., Large sample theory of intrinsic and extrinsic sample means on manifold (part i), Ann. statist., 31, 1, 1-29, (2003) · Zbl 1020.62026
[5] Boothby, W., An introduction to differentiable manifolds and Riemannian geometry, (1975), Academic Press New York · Zbl 0333.53001
[6] Chavel, I., 1993. Riemannian geometry: a modern introduction. In: Cambridge Tracts in Mathematics, vol. 108. Cambridge University Press, Cambridge. · Zbl 0810.53001
[7] Corcuera, J.; Kendall, W., Riemannian barycentres and geodesic convexity, Math. proc. Cambridge philos. soc., 127, 2, 253-269, (1999) · Zbl 0948.53020
[8] Devroye, L.; Gyorfi, L., Nonparametric density estimation. the \(L_1\)-view, (1985), Wiley New York · Zbl 0546.62015
[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. · Zbl 0753.60046
[10] Fischer, N.; Lewis, T.; Embleton, B., Statistical analysis of spherical data, (1993), Cambridge University Press Cambridge · Zbl 0782.62059
[11] Hall, P.; Watson, G.; Cabrera, J., Kernel density estimation with spherical data, Biometrika, 74, 751-762, (1987) · Zbl 0632.62033
[12] Healy, D.; Kim, P., An empirical Bayes approach to directional data and efficient computation on the sphere, Ann. statist., 24, 232-254, (1996) · Zbl 0856.62010
[13] Healy, D.; Hendriks, H.; Kim, P., Spherical deconvolution, J. multivariate anal., 67, 1-22, (1998) · Zbl 1126.62346
[14] Hebey, E., Introduction à l’analyse non linéaire sur LES variétés, (1997), Diderot Paris, New York · Zbl 0918.58001
[15] Hendriks, H., Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions, Ann. statist., 18, 832-849, (1990) · Zbl 0711.62036
[16] Hendriks, H., Application of fast spherical Fourier transform to density estimation, J. multivariate anal., 84, 209-221, (2003) · Zbl 1026.62033
[17] Hendriks, H.; Janssen, J.; Ruymgaart, F., Strong uniform convergence of density estimators on compact Euclidean manifolds, Statist. probab. lett., 16, 305-311, (1993) · Zbl 0766.62020
[18] Jupp, P.; Mardia, K., A unified view of the theory of directional statistics 1975-1988, Internat. statist. rev., 57, 261-294, (1989) · Zbl 0707.62095
[19] Karcher, H., Riemannian center of mass and mollifier smoothing, Comm. pure appl. math., 30, 509-554, (1977) · Zbl 0354.57005
[20] Kobayashi, S., Nomizu, K., 1969. Foundations of Differential Geometry, vols. 1,2. Wiley, New York. · Zbl 0175.48504
[21] Le, H., On the consistency of procrustean Mean shapes, Adv. appl. probab., 30, 53-63, (1998) · Zbl 0906.60007
[22] Lee, J.; Ruymgaart, F., Nonparametric curve estimation on Stiefel manifolds, Nonparam. statist., 6, 57-68, (1996) · Zbl 0862.62037
[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] Mardia, K., Statistics of directional data, (1972), Academic Press New York · Zbl 0244.62005
[25] Oller, J.; Corcuera, J., Intrinsic analysis of statistical estimation, Ann. statist., 23, 5, 1562-1581, (1995) · Zbl 0843.62027
[26] Parzen, E., On the estimation of a probability density function and the mode, Ann. math. statist., 33, 1065-1076, (1962) · Zbl 0116.11302
[27] Pennec, X.; Ayache, N., Uniform distribution, distance and expectation problems for geometric features processing, J. math. imaging vision, 9, 1, 49-67, (1998) · Zbl 0906.68198
[28] Rosenblatt, M., Remarks on some nonparametric estimates of a density function, Ann. math. statist., 27, 832-837, (1956) · Zbl 0073.14602
[29] Van der Vaart, A., Asymptotic statistics, (1998), Cambridge University Press Cambridge · Zbl 0910.62001
[30] Watson, G., Statistics on spheres, (1983), Wiley New York · Zbl 0646.62045
[31] Willmore, T., Riemannian geometry, (1993), Oxford University Press Oxford · Zbl 0797.53002
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