We apply the holonomic gradient method introduced by Nakayama et al. (2011) [23] to the evaluation of the exact distribution function of the largest root of a Wishart matrix, which involves a hypergeometric function 1F1 of a matrix argument. Numerical evaluation of the hypergeometric function...

We study properties of Fisher distribution (von Mises–Fisher distribution, matrix Langevin distribution) on the rotation group SO(3). In particular we apply the holonomic gradient descent, introduced by Nakayama et al. (2011) [16], and a method of series expansion for evaluating the...

Markov basis for statistical model of contingency tables gives a useful tool for performing the conditional test of the model via the Markov chain Monte Carlo method. In this paper, we derive explicit forms of Markov bases for change point models and block diagonal effect models, which are...

For the purpose of testing the spherical uniformity based on i.i.d. directional data (unit vectors) zi, i=1,...,n, Anderson and Stephens (Biometrika 59 (1972) 613-621) proposed testing procedures based on the statistics Smax=maxu S(u) and Smin=minu S(u), where u is a unit vector and nS(u) is the...

We consider the asymptotic joint distribution of the eigenvalues and eigenvectors of Wishart matrix when the population eigenvalues become infinitely dispersed. We show that the normalized sample eigenvalues and the relevant elements of the sample eigenvectors are asymptotically all mutually...

We give James-Stein type estimators of a multivariate normal mean vector by shrinkage towards a closed convex set K with a smooth or piecewise smooth boundary. The rate of shrinkage is determined by the curvature of the boundary of K at the projection point onto K. By considering a sequence of...

For orthogonally invariant estimation of [Sigma] of Wishart distribution using Stein's loss, any estimator which does not preserve the order of the sample eigenvalues is dominated by a modified estimator preserving the order.

This paper deals with the asymptotic distribution of Wishart matrix and its application to the estimation of the population matrix parameter when the population eigenvalues are block-wise infinitely dispersed. We show that the appropriately normalized eigenvectors and eigenvalues asymptotically...

Pairwise linear discriminant analysis ofmpopulations inRncan be regarded as a process to generate rankings of the populations. Whennis small compared withm, some of them! rankings are not generated. We give formulae for the number of generated rankings. Moreover, we give basic characterizations...

We consider the numerical evaluation of one-dimensional projections of general multivariate stable densities introduced by Abdul-Hamid and Nolan [H. Abdul-Hamid, J.P. Nolan, Multivariate stable densities as functions of one dimensional projections, J. Multivariate Anal. 67 (1998) 80-89]. In...