Deconvolution density estimation on SO(N)

This paper develops nonparametric deconvolution density estimation over SO(N), the group of N x N orthogonal matrices of determinant 1. The methodology is to use the group and manifold structures to adapt the Euclidean deconvolution techniques to this Lie group environment. This is achieved by employing the theory of group representations explicit to SO(N), General consistency results are obtained with specific rates of convergence achieved under sufficient smoothness conditions. Application to empirical Bayes prior estimation and inference is also discussed.

[1]  D. Flinn.  Orientation Statistics , 1967, Nature.

[2]  N. Vilenkin.  Special Functions and the Theory of Group Representations , 1968 .

[3]  Eugene P. Wigner,et al.  Special Functions. A Group Theoretic Approach , 1969 .

[4]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[5]  M. Giné,et al.  Invariant Tests for Uniformity on Compact Riemannian Manifolds Based on Sobolev Norms , 1975 .

[6]  A. Feuerverger,et al.  The Empirical Characteristic Function and Its Applications , 1977 .

[7]  Jean-Pierre Serre,et al.  Linear representations of finite groups , 1977, Graduate texts in mathematics.

[8]  K. Mardia,et al.  The von Mises–Fisher Matrix Distribution in Orientation Statistics , 1977 .

[9]  S. Helgason.  Differential Geometry, Lie Groups, and Symmetric Spaces , 1978 .

[10]  Rudolf Beran,et al.  Exponential Models for Directional Data , 1979 .

[11]  Prakasa Rao.  Nonparametric functional estimation , 1983 .

[12]  P. E. Jupp,et al.  Sobolev Tests for Symmetry of Directional Data , 1983 .

[13]  S. Helgason.  Groups and geometric analysis , 1984 .

[14]  Luc Devroye,et al.  Nonparametric Density Estimation , 1985 .

[15]  L. Devroye,et al.  Nonparametric Density Estimation: The L 1 View. , 1985 .

[16]  Tammo tom Dieck,et al.  Representations of Compact Lie Groups , 1985 .

[17]  R. H. Farrell,et al.  Multivariate calculation : use of the continuous groups , 1986 .

[18]  M. J. Prentice.  Orientation Statistics Without Parametric Assumptions , 1986 .

[19]  P. Hall,et al.  Kernel density estimation with spherical data , 1987 .

[20]  C. D. Kemp,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[21]  Calyampudi R. Rao,et al.  Kernel estimators of density function of directional data , 1988 .

[22]  P. Diaconis.  Group representations in probability and statistics , 1988 .

[23]  C. W. Langenberg,et al.  Statistical classification of macroscopic folds as cylindrical, circular conical, or elliptical conical , 1988 .

[24]  J. Lo,et al.  Characterizing Fourier series representation of probability distributions on compact Lie groups , 1988 .

[25]  Donald St. P. Richards,et al.  Analogs and Extensions of Selberg’s Integral , 1989 .

[26]  Jianqing Fan,et al.  Global Behavior of Deconvolution Kernel Estimates , 1989 .

[27]  L. Devroye.  Consistent deconvolution in density estimation , 1989 .

[28]  Michael J. Prentice,et al.  Spherical Regression on Matched Pairs of Orientation Statistics , 1989 .

[29]  Ted Chang,et al.  Spherical regression with errors in variables , 1989 .

[30]  Cun-Hui Zhang.  Fourier Methods for Estimating Mixing Densities and Distributions , 1990 .

[31]  Daniel Q. Naiman,et al.  Volumes of Tubular Neighborhoods of Spherical Polyhedra and Statistical Inference , 1990 .

[32]  Dennis V. Lindley,et al.  Empirical Bayes Methods , 1974 .

[33]  Harrie Hendriks,et al.  Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions , 1990 .

[34]  P. Kim.  Decision theoretic analysis of spherical regression , 1991 .

[35]  Jianqing Fan.  On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems , 1991 .

[36]  Gong Sheng,et al.  Harmonic Analysis on Classical Groups , 1991 .

[37]  J. S. Maritz,et al.  Empirical Bayes Methods , 1974 .

[38]  David Keith Maslen.  Fast transforms and sampling for compact groups , 1993 .

[39]  Peter J. Diggle,et al.  A Fourier Approach to Nonparametric Deconvolution of a Density Estimate , 1993 .

[40]  Jianqing Fan,et al.  Nonparametric regression with errors in variables , 1993 .

[41]  J. Rosenthal.  Random Rotations: Characters and Random Walks on SO(N) , 1994 .

[42]  D. Healy,et al.  Computing Fourier Transforms and Convolutions on the 2-Sphere , 1994 .

[43]  Kanti V. Mardia,et al.  Shape changes in the plane for landmark data , 1995 .

[44]  B. Lindsay.  Mixture models : theory, geometry, and applications , 1995 .

[45]  Dennis M. Healy,et al.  An empirical Bayes approach to directional data and efficient computation on the sphere , 1996 .

[46]  K. Maurin.  Representations of Compact Lie Groups , 1997 .

[47]  Dennis M. Healy,et al.  Spherical Deconvolution , 1998 .