An empirical Bayes approach to directional data and efficient computation on the sphere

Annals of Statistics

Volumn 24, Issue 1, 1996, Pages 232-254

  Healy Jr , Dennis M ^a   Kim, Peter T ^b  

^a DARTMOUTH COLLEGE   (United States)

^b UNIVERSITY OF GUELPH   (Canada)

Author keywords

Consistency; Fast Fourier transform; Legendre transform; Spherical harmonics

Indexed keywords

EID: 0030543857     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/aos/1033066208     Document Type: Article

References (33)

1. BAGCHI, P. and GUTTMAN, I. (1988). Theoretical considerations of the multivariate von Mises-Fisher distribution. Journal of Applied Statistics 15 149-169.
2. BAI, Z. D., RAO, C. R. and ZHAO, L. C. (1988). Kernel estimators of density function of directional data. J. Multivariate Anal. 27 24-39.
3. BERAN, R. (1979). Exponential models for directional data. Ann. Statist. 7 1162-1178.
4. BRUNNER, L. J. and Lo, A. Y. (1989). Bayes methods for a symmetric unimodal density and its mode. Ann. Statist. 17 1550-1566
5. COURANT, R. and HILBERT, D. (1953). Methods of Mathematical Physics 1. Interscience, New York.
6. DIACONIS, P. (1988). Group Representations in Probability and Statistics. IMS, Hayward, CA.
7. DRISCOLL, J. R. and HEALY, D. M., JR. (1994). Computing Fourier transforms and convolutions on the 2-sphere. Adv. in Appl. Math. 15 202-250.
8. FAN, J. (1991). Global behavior of deconvolution kernel estimates. Statist. Sinica 1 541-551.
9. FERGUSON, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209-230.
10. FISHER, N. I., LEWIS, T. and EMBLETON, B. J. (1987). Statistical Analysis of Spherical Data. Cambridge Univ. Press.
11. FISHER, R. A. (1953). Dispersion on a sphere. Proc. Roy. Soc. London Ser. A 217 295-305.
12. GINÉ, E. (1975). Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. Ann. Statist. 3 1243-1266.
13. GRUNBAUM, F. A., LONGHI, L. and PERLSTADT, M. (1982). Differential operators commuting with finite convolution integral operators: some non-abelian examples. SIAM J. Appl. Math. 42 941-955.
14. GUTTORP, P. and LOCKHART, R. A. (1988). Finding the location of a signal: A Bayesian analysis. J. Amer. Statist. Assoc. 83 322-330.
15. HALL, P., WATSON, G. and CABRERA, J. (1987). Kernel density estimation with spherical data. Biometrika 74 751-762.
16. HEALY, D., MOORE, S. and ROCKMORE, D. (1993). Efficiency and stability issues in the numerical computation of Fourier transforms and convolutions on the 2-sphere. Preprint, Dartmouth College.
17. HELGASON, S. (1984). Groups and Geometric Analysis. Academic Press, New York.
18. HENDRIKS, H. (1990). Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions. Ann. Statist. 18 832-849.
19. JUPP, P. E. and MARDIA, K. V. (1989). A unified view of the theory of directional statistics, 1975-1988. Internat. Statist. Rev. 57 261-294.
20. LO, A. Y. and CABRERA, J. (1987). Bayes procedures for rotationally symmetric models on the sphere. Ann. Statist. 15 1257-1263.
21. LO, J. T. and ESHELMAN, L. R. (1979). Exponential Fourier densities on SO(S) and optimal estimation and detection for rotational processes. SIAM J. Appl. Math. 36 73-82.
22. MARDIA, K. V. (1972). Statistics of Directional Data. Academic Press, New York.
23. MARDIA, K. V. (1975). Statistics of directional data (with discussion). J. Roy. Statist. Soc. Ser. B 37 349-393.
24. MARDIA, K. V. and EL-ATOUM, S. A. M. (1976). Bayesian inference for the von Mises-Fisher distribution. Biometrika 63 203-206.
25. MASLEN, D. (1993). Fast transforms and sampling for compact groups. Ph.D. Dissertation, Dept. Mathematics, Harvard Univ.
26. ROBBINS, H. (1955). An empirical Bayes approach to statistics. Proc. Third Berkeley Symp. Math. Statist. 1 157-164. Univ. California Press, Berkeley.
27. SILVERMAN, B. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.
28. SINGH, R. S. (1992). Empirical Bayes with rates and best rates of convergence in u(x)C(θ)exp(-x/θ)-family: Estimation case. Ann. Inst. Statist. Math. 44 435-449.
29. TALMAN, J. D. (1968). Special Functions: A Group Theoretic Approach. W. A. Benjamin, New York.
30. VILENKIN, N. J. (1968). Special Functions and the Theory of Group Representations. Amer. Math. Soc., Providence, RI.
31. WAHBA, G. (1981). Spline interpolation and smoothing on the sphere. SIAM J. Sci. Statist. Comput. 2 5-16.
32. WATSON, G. S. (1983). Statistics on Spheres. Wiley, New York.
33. ZHANG, C. H. (1990). Fourier methods for estimating mixing densities and distributions. Ann. Statist. 18 806-831.