Angers, Jean-François; Kim, Peter T. Multivariate Bayesian function estimation. (English) Zbl 1084.62032 Ann. Stat. 33, No. 6, 2967-2999 (2005). Summary: Bayesian methods are developed for the multivariate nonparametric regression problem where the domain is taken to be a compact Riemannian manifold. In terms of the latter, the underlying geometry of the manifold induces certain symmetries on the multivariate nonparametric regression function. The Bayesian approach then allows one to incorporate hierarchical Bayesian methods directly into the spectral structure, thus providing a symmetry-adaptive multivariate Bayesian function estimator. One can also diffuse away some prior information in which the limiting case is a smoothing spline on the manifold. This, together with the result that the smoothing spline solution obtains the minimax rate of convergence in the multivariate nonparametric regression problem, provides good frequentist properties for the Bayes estimators. An application to astronomy is included. Cited in 5 Documents MSC: 62G08 Nonparametric regression and quantile regression 62C10 Bayesian problems; characterization of Bayes procedures 58A35 Stratified sets 41A15 Spline approximation 58J90 Applications of PDEs on manifolds Keywords:Bayes factor; comets; cross-validation; eigenfunctions; eigenvalues; posterior; Sobolev spaces; zeta function × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Angers, J.-F. and Delampady, M. (1992). Hierarchical Bayesian curve fitting and smoothing. Canad. J. Statist. 20 35–49. JSTOR: · Zbl 0761.62029 · doi:10.2307/3315573 [2] Angers, J.-F. and Delampady, M. (1997). Hierarchical Bayesian curve fitting and model choice for spatial data. Sankhyā Ser. B 59 28–43. · Zbl 0919.62027 [3] Angers, J.-F. and Delampady, M. (2001). Bayesian nonparametric regression using wavelets. Sankhyā Ser. A 63 287–308. · Zbl 1192.62111 [4] Beran, R. J. (1968). Testing for uniformity on a compact homogeneous space. J. Appl. Probability 5 177–195. JSTOR: · Zbl 0174.50203 · doi:10.2307/3212085 [5] Beran, R. (1979). Exponential models for directional data. Ann. Statist. 7 1162–1178. · Zbl 0426.62030 · doi:10.1214/aos/1176344838 [6] Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis , 2nd ed. Springer, New York. · Zbl 0572.62008 [7] Chavel, I. (1984). Eigenvalues in Riemannian Geometry . Academic Press, Orlando, FL. · Zbl 0551.53001 [8] Chikuse, Y. (2003). Statistics on Special Manifolds . Lecture Notes in Statist. 174 . Springer, New York. · Zbl 1026.62051 [9] Chikuse, Y. and Jupp, P. E. (2004). A test of uniformity on shape spaces. J. Multivariate Anal. 88 163–176. · Zbl 1041.62047 · doi:10.1016/S0047-259X(03)00066-6 [10] Chirikjian, G. S. and Kyatkin, A. B. (2001). Engineering Applications of Noncommutative Harmonic Analysis . CRC Press, Boca Raton, FL. · Zbl 1100.42500 [11] Cox, D. D. (1984). Multivariate smoothing spline functions. SIAM J. Numer. Anal. 21 789–813. JSTOR: · Zbl 0581.65012 · doi:10.1137/0721053 [12] Cox, D. D. (1988). Approximation of method of regularization estimators. Ann. Statist. 16 694–712. · Zbl 0671.62044 · doi:10.1214/aos/1176350829 [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. · Zbl 0488.65079 · doi:10.1137/0513010 [14] Efromovich, S. (2000). On sharp adaptive estimation of multivariate curves. Math. Methods Statist. 9 117–139. · Zbl 1006.62033 [15] Fisher, N. I., Lewis, T. and Embleton, B. J. J. (1993). Statistical Analysis of Spherical Data . Cambridge Univ. Press. · Zbl 0782.62059 [16] Giné, E. M. (1975). Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. Ann. Statist. 3 1243–1266. · Zbl 0322.62058 · doi:10.1214/aos/1176343283 [17] Hanna, M. S. and Chang, T. (2000). Fitting smooth histories to rotation data. J. Multivariate Anal. 75 47–61. · Zbl 0980.62109 · doi:10.1006/jmva.2000.1893 [18] Healy, D. M., Hendriks, H. and Kim, P. T. (1998). Spherical deconvolution. J. Multivariate Anal. 67 1–22. · Zbl 1126.62346 · doi:10.1006/jmva.1998.1757 [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. · Zbl 0856.62010 · doi:10.1214/aos/1033066208 [20] Hendriks, H. (1990). Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions. Ann. Statist. 18 832–849. · Zbl 0711.62036 · doi:10.1214/aos/1176347628 [21] Jupp, P. E. and Kent, J. T. (1987). Fitting smooth paths to spherical data. Appl. Statist. 36 34–46. JSTOR: · Zbl 0613.62086 · doi:10.2307/2347843 [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. · Zbl 1039.85002 · doi:10.1198/016214503000000305 [23] Jupp, P. E. and Spurr, B. D. (1983). Sobolev tests for symmetry of directional data. Ann. Statist. 11 1225–1231. · Zbl 0551.62035 [24] Kim, P. T. (1998). Deconvolution density estimation on \(\mathitSO(N)\). Ann. Statist. 26 1083–1102. · Zbl 0929.62042 · doi:10.1214/aos/1024691089 [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. JSTOR: · Zbl 0981.62032 · doi:10.2307/3315986 [26] Kim, P. T. and Koo, J.-Y. (2002). Optimal spherical deconvolution. J. Multivariate Anal. 80 21–42. · Zbl 0998.62030 · doi:10.1006/jmva.2000.1968 [27] Kim, P. T. and Richards, D. St. 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. · Zbl 1020.62029 [28] Klemelä, J. (1999). Asymptotic minimax risk for the white noise model on the sphere. Scand. J. Statist. 26 465–473. · Zbl 0939.62092 · doi:10.1111/1467-9469.00160 [29] Lee, J. and Ruymgaart, F. H. (1996). Nonparametric curve estimation on Stiefel manifolds. J. Nonparametr. Statist. 6 57–68. · Zbl 0862.62037 · doi:10.1080/10485259608832663 [30] 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. JSTOR: · Zbl 0246.62050 [31] Luo, Z. (1998). Backfitting in smoothing spline ANOVA. Ann. Statist. 26 1733–1759. · Zbl 0929.62043 · doi:10.1214/aos/1024691355 [32] Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics . Wiley, Chichester. · Zbl 0935.62065 [33] Marsden, B. G. and Williams, G. V. (1993). Catalogue of Cometary Orbits , 8th ed. Minor Planet Center, Smithsonian Astrophysical Observatory, Cambridge, MA. [34] Minakshisundaram, S. and Pleijel, Å. (1949). Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Canadian J. Math. 1 242–256. · Zbl 0041.42701 · doi:10.4153/CJM-1949-021-5 [35] Pinsker, M. S. (1980). Optimal filtering of square integrable signals in Gaussian white noise. Problems Inform. Transmission 16 (1) 52–68. · Zbl 0452.94003 [36] Prentice, M. J. (1987). Fitting smooth paths to rotation data. Appl. Statist. 36 325–331. JSTOR: · doi:10.2307/2347791 [37] Robert, C. P. (2001). The Bayesian Choice , 2nd ed. Springer, New York. · Zbl 0980.62005 [38] Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461–464. · Zbl 0379.62005 · doi:10.1214/aos/1176344136 [39] Searle, S. R. (1982). Matrix Algebra Useful for Statistics . Wiley, New York. · Zbl 0555.62002 [40] Speckman, P. (1985). Spline smoothing and optimal rates of convergence in nonparametric regression models. Ann. Statist. 13 970–983. · Zbl 0585.62074 · doi:10.1214/aos/1176349650 [41] Taijeron, H., Gibson, A. and Chandler, C. (1994). Spline interpolation and smoothing on hyperspheres. SIAM J. Sci. Comput. 15 1111–1125. · Zbl 0812.41009 · doi:10.1137/0915068 [42] Turski, J. (1998). Harmonic analysis on \(SL(2,\C)\) and projectively adapted pattern representation. J. Fourier Anal. Appl. 4 67–91. · Zbl 0911.43004 · doi:10.1007/BF02475928 [43] Turski, J. (2000). Projective Fourier analysis for patterns. Pattern Recognition 33 2033–2043. · Zbl 1006.68842 · doi:10.1016/S0031-3203(99)00196-X [44] 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. · Zbl 0737.62041 [45] Wahba, G. (1981). Spline interpolation and smoothing on the sphere. SIAM J. Sci. Statist. Comput. 2 5–16. [Erratum (1982) 3 385–386.] · Zbl 0537.65008 · doi:10.1137/0902002 [46] Wahba, G. (1990). Spline Models for Observational Data . SIAM, Philadelphia. · Zbl 0813.62001 [47] Watson, G. S. (1983). Statistics on Spheres . Wiley, New York. · Zbl 0646.62045 [48] Wellner, J. A. (1979). Permutation tests for directional data. Ann. Statist. 7 929–943. · Zbl 0417.62044 · doi:10.1214/aos/1176344779 [49] Wiegert, P. and Tremaine, S. (1999). The evolution of long-period comets. Icarus 137 84–121. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.