数理解析研究所講究録 1308 巻 2003 年 112-113

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Some Results in Asymptotic Theory on Special Manifolds
筑瀬靖子 (Yasuko Chikuse)

香川大・工学部

Faculty of Engineering, Kagawa University

1Introduction
We may be interested in statistical analysis on special manifolds, e.g., Stiefel,
Grassmann and shape manifolds. The analysis of data on these manifolds is required
in practical applications in many fields ;the Grassmann manifold is arather new
subject treated as asample space.
We usually consider, as population distributions, exponential-type distributions
on the manifolds; e.g., the matrix Langevin and the matrix Bingham distributions
including the uniform distributions. These distributions are relatively tractable likethe normal distributions on the Euclidean space. However, the distribution forms andinference problems based on these exponential distributions involve hypergeometricfunctions with matrix arguments and the exact solutions are expressed in integral ordifferential forms involving these hypergeometric functions. Thus we need to evaluate
these problems asymptotically, that is, for large sample size (and small concentration),
for large concentration, and for high dimension.
We concentrate our discussion on the Stiefel manifold (and the Grassmann manifold as well if time is permitted). The Stiefel manifold
is the space apointof which is aset of orthonormal vectors in $H^{n}(k \leq m)$ , so that $V_{k,m}=\{X(m\cross$$k);X’X=I_{k}\}$ , whereis theidentity matrix. Special cases are the unit
hypersphereof directed vectors and the orthogonal group $O(m)=V_{m,m}$ of$m\cross m$ orthonormal matrices. The analysis ofdata on
is required especially for$\leq m\leq 3$ in practical applications in Geological Sciences,Astrophysics, Biology,Meteorology, Medicine and other fields.
The matrix Langevin $L(m, k;F)$ distribution is of exponential type whose density
function is given by$V_{k,m}$

$k$

$k$

$I_{k}$$\cross k$$V_{1,m}$$V_{k,m}$$k$$\exp(trF’X)/_{0}F_{1}.(\frac{1}{2}m;\frac{1}{4}F’F)$

for an$m\mathrm{x}k$$F=\Gamma\Lambda\Theta’$,matrix $F$ of rank $p(\leq k)$ with the singular value decomposition of $F$, where$\Gamma\in V_{p,m}$,$\Theta\in V_{p,k}$, and$\mathrm{A}=diag(\lambda_{1}, \ldots, \lambda_{p})$,$\lambda_{1}\geq\cdots\geq\lambda_{p}>0$,(see Downs (1972)). Some more distributions of the $L(m, k; F)$ distribution may befound in e.g., Jupp and Mardia (1979), Watson (1983), Prentice (1986), and Chikuse$(1998, 1999, 2002\mathrm{b})$.

2Asymptotic Results
Large sample asymptotic theory is concerned in connection with tests for uniformity (A $=0$ ) of the $L(m, k;F)$ distribution (see Chikuse (1991)). Asymptotic

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properties, near the uniformity, are discussed of the estimation of the orientation
parameters and and the concentration parameter , and of some optimal tests
for uniformity (the likelihoo ratio, the locally best invariant, the Rao score and theRayleigh-style tests).
Asymptotic theory is developed for the concentrated $L(m, k;F)$ distributions (i.e.,for large ), discussing the estimation of the large concentration parameter , tests for
hypotheses of the orientation parameters and $$ , classifications of matrix Langevindistributions, measures of orthogonal association and orientational regressions (seee.g., Chikuse (2000a) .
This papar investigates, in particular, the high dimensional asymptotic behaviorof some matrix statistics and related functions constructed from some main distributions, including the matrix Langevin, the matrix Bingham and the uniform distributions. We derive asymptotic expantions for the distributions of the standardized
sample mean matrices, using the results on the invariant polynomials and the generalized Hermite and Laguerre polynomials with matrix arguments.
We generalize the Stam’s (first and second) limit theorems, for high dimensional
samples taken from the hypersphere, to some non-uniform distributions on the Stiefel
manifold. We discuss high dimensional asymptotic properties of the parameter estimation and the tests of hypotheses for the $L(m, k;F)$ distributions. We are also
concerned with large sample asymptotic properties of the inference based on the(profile) score functions when the dimension is large.
$\Gamma$$\Lambda$$\Theta$$\Lambda$$\Lambda$$\Gamma$$)$REFERENCES
Chikuse, Y. (1991). Asymptotic expansions for distributions of the large sample
matrix resultant and related statistics on the Stiefel manifold. Journal of Multivariate Analysis 39, 270-283.
Chikuse, Y. (1998). Density estimation on the Stiefel manifold. Journal of Multivariate Analysis 66, 188-206.
Chikuse, Y. (1999). Procrustes analysis on some special manifolds. Communicationsin Statistics- Theory and Methods 28, 885-903.
Chikuse, Y. (2002a). Concentrated matrix Langevin distributions. To appear in
Journal of Multivariate Analysis.
Chikuse, Y. (2002b). Statistics on Special Manifolds.
Downs, T.D. (1972). Orientation statistics. Biometrica 59, 665-676.
Jupp, P.E. and Mardia, K.V. (1979). Maximum likelihood estimators for the matrix
von Mises-Fisher and Bingham distributions. Annals of Statistics 7, 599-606.
Prentice, M.J. (1986). Orientation statistics without parametric assumptions. Journal of Royal Statistical Society Series B48,214-222.
Stam, A.J. (1982). Limit theorems for uniform distributions on spheres in highdimensional euclidean spaces. J. App. Prob. 19, 221-228.
Watson, G.S. (1983). Statistics on Spheres. Lecture Note in Mathematics (University of Arkansas) 6, Wiley. New York