Multivariate bayesian function estimation

Annals of Statistics

Volumn 33, Issue 6, 2005, Pages 2967-2999

  Angers, Jean François ^a   Kim, Peter T ^b  

^a UNIVERSITÉ DE MONTRÉAL   (Canada)

^b UNIVERSITY OF GUELPH   (Canada)

Author keywords

Bayes factor; Comets; Cross validation; Eigenfunctions; Eigenvalues; Posterior; Sobolev spaces; Zeta function

Indexed keywords

EID: 33644921211     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053605000000705     Document Type: Article

References (48)

1. ANGERS, J.-F. and DELAMPADY, M. (1992). Hierarchical Bayesian curve fitting and smoothing. Canad. J. Statist. 20 35-49. MR1173059
2. ANGERS, J.-F. and DELAMPADY, M. (1997). Hierarchical Bayesian curve fitting and model choice for spatial data. Sankhyā Ser. B 59 28-43. MR1733376
3. ANGERS, J.-F. and DELAMPADY, M. (2001). Bayesian nonparametric regression using wavelets. Sankhyā Ser. A 63 287-308. MR 1897044
4. BERAN, R. J. (1968). Testing for uniformity on a compact homogeneous space. J. Appl. Probability 5 177-195. MR0228098
5. BERAN, R. (1979). Exponential models for directional data. Ann. Statist. 7 1162-1178. MR0550142
6. BERGER, J. O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer, New York. MR0804611
7. CHAVEL, I. (1984). Eigenvalues in Riemannian Geometry. Academic Press, Orlando, FL. MR0768584
8. CHIKUSE, Y. (2003). Statistics on Special Manifolds. Lecture Notes in Statist. 174. Springer, New York. MR 1960435
9. CHIKUSE, Y. and JUPP, P. E. (2004). A test of uniformity on shape spaces. J. Multivariate Anal. 88 163-176. MR2021868
10. CHIRIKJIAN, G. S. and KYATKIN, A. B. (2001). Engineering Applications of Noncommutative Harmonic Analysis. CRC Press, Boca Raton, FL. MR1885369
11. COX, D. D. (1984). Multivariate smoothing spline functions. SIAM J. Numer. Anal. 21789-813. MR0749371
12. Cox, D. D. (1988). Approximation of method of regularization estimators. Ann. Statist. 16 694-712. MR0947571
13. DYN, N. and WAHBA, G. (1982). On the estimation of functions of several variables from aggregated data. SIAM J. Math. Anal. 13 134-152. MR0641546
14. EFROMOVICH, S. (2000). On sharp adaptive estimation of multivariate curves. Math. Methods Statist. 9 117-139. MR1780750
15. FISHER, N. I., LEWIS, T. and EMBLETON, B. J. J. (1993). Statistical Analysis of Spherical Data. Cambridge Univ. Press. MR1247695
16. GINÉ, E. M. (1975). Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. Ann. Statist. 3 1243-1266. MR0388663
17. HANNA, M. S. and CHANG, T. (2000). Fitting smooth histories to rotation data. J. Multivariate Anal. 75 47-61. MR1787401
18. HEALY, D. M., HENDRIKS, H. and KIM, P. T. (1998). Spherical deconvolution. J. Multivariate Anal. 67 1-22. MR1659108
19. HEALY, D. M. and KIM, P. T. (1996). An empirical Bayes approach to directional data and efficient computation on the sphere. Ann. Statist. 24 232-254. MR1389889
20. HENDRIKS, H. (1990). Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions. Ann. Statist. 18 832-849. MR1056339
21. JUPP, P. E. and KENT, J. T. (1987). Fitting smooth paths to spherical data. Appl. Statist. 36 34-46. MR0887825
22. JUPP, P. E., KIM, P. T., KOO, J.-Y. and WIEGERT, P. (2003). The intrinsic distribution and selection bias of long-period cometary orbits. J. Amer. Statist. Assoc. 98 515-521. MR2011668
23. JUPP, P. E. and SPURR, B. D. (1983). Sobolev tests for symmetry of directional data. Ann. Statist. 11 1225-1231. MR0720267
24. KIM, P. T. (1998). Deconvolution density estimation on SO(N). Ann. Statist. 26 1083-1102. MR1635446
25. KIM, P. T. and KOO, J.-Y. (2000). Directional mixture models and optimal estimation of the mixing density. Canad. J. Statist. 28 383-398. MR1792056
26. KIM, P. T. and KOO, J.-Y. (2002). Optimal spherical deconvolution. J. Multivariate Anal. 80 21-42. MR1889831
27. KIM, P. T. and RICHARDS, D. S. P. (2001). Deconvolution density estimation on compact Lie groups. In Algebraic Methods in Statistics and Probability (M. A. G. Viana and D. St. P. Richards, eds.) 155-171. Amer. Math. Soc., Providence, RI. MR1873674
28. KLEMELÄ, J. (1999). Asymptotic minimax risk for the white noise model on the sphere. Scand. J. Statist. 26465-473. MR1712031
29. LINDLEY, D. V. and SMITH, A. F. M. (1972). Bayes estimates for the linear model (with discussion). J. Roy. Statist. Soc. Ser. B 34 1-41. MR0415861
30. Luo, Z. (1998). Backfitting in smoothing spline ANOVA. Ann. Statist. 26 1733-1759. MR1673276
31. MARDIA, K. V. and JUPP, P. E. (2000). Directional Statistics. Wiley, Chichesler. MR1828667
32. MARSDEN, B. G. and WILLIAMS, G. V. (1993). Catalogue of Cometary Orbits, 8th ed. Minor Planet Center, Smithsonian Astrophysical Observatory, Cambridge, MA.
33. MINAKSHISUNDARAM, S. and PLEIJEL, Å. (1949). Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Canadian J. Math. 1 242-256. MR0031145
34. PINSKER, M. S. (1980). Optimal filtering of square integrable signals in Gaussian white noise. Problems Inform. Transmission 16 (1) 52-68. MR0624591
35. PRENTICE, M. J. (1987). Fitting smooth paths to rotation data. Appl. Statist. 36 325-331. MR0918853
36. ROBERT, C. P. (2001). The Bayesian Choice, 2nd ed. Springer, New York. MR1835885
37. SCHWARZ, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461-464. MR0468014
38. SEARLE, S. R. (1982). Matrix Algebra Useful for Statistics. Wiley, New York. MR0670947
39. SPECKMAN, P. (1985). Spline smoothing and optimal rates of convergence in nonparametric regression models. Ann. Statist. 13 970-983. MR0803752
40. TAIJERON, H., GIBSON, A. and CHANDLER, C. (1994). Spline interpolation and smoothing on hyperspheres. SIAM J. Sci. Comput. 15 1111-1125. MR1289156
41. TURSKI, J. (1998). Harmonic analysis on SL(2, ℂ) and projectively adapted pattern representation. J. Fourier Anal. Appl. 4 67-91. MR1650956
42. TURSKI, J. (2000). Projective Fourier analysis for patterns. Pattern Recognition 33 2033-2043.
43. VAN ROOIJ, A. C. M. and RUYMGAART, F. H. (1991). Regularized deconvolution on the circle and the sphere. In Nonparametric Functional Estimation and Related Topics (G. Roussas, ed.) 679-690. Kluwer, Dordrecht. MR1154359
44. WAHBA, G. (1981). Spline interpolation and smoothing on the sphere. SIAM J. Sci. Statist. Comput. 2 5-16. [Erratum (1982) 3 385-386.] MR0618629
45. WAHBA, G. (1990). Spline Models for Observational Data. SIAM, Philadelphia. MR1045442
46. WATSON, G. S. (1983). Statistics on Spheres. Wiley, New York. MR0709262
47. WELLNER, J. A. (1979). Permutation tests for directional data. Ann. Statist. 7 929-943. MR0536498
48. WIEGERT, P. and TREMAINE, S. (1999). The evolution of long-period comets. Icarus 137 84-121.