Yasuko Chikuse

A Survey on the Invariant Polynomials with Matrix Arguments in Relation to Econometric Distribution Theory

Invariant polynomials with matrix arguments have been defined by the theory of group representations, generalizing the zonal polynomials. They have developed as a useful tool to evaluate certain integrals arising in multivariate distribution theory, which were expanded as power series in terms of the invariant polynomials. Some interest in the polynomials has been shown by people working in the field of econometric theory. In this paper, we shall survey the properties of the invariant polynomials and their applications in multivariate distribution theory including related developments in econometrics.

Some properties of invariant polynomials with matrix arguments and their applications in econometrics

SummaryFurther properties are derived for a class of invariant polynomials with several matrix arguments which extend the zonal polynomials. Generalized Laguerre polynomials are defined, and used to obtain expansions of the sum of independent noncentral Wishart matrices and an associated generalized regression coefficient matrix. The latter includes thek-class estimator in econometrics.

Methods for Constructing Top Order Invariant Polynomials

The invariant polynomials null (Davis [8] and Chikuse [2] with r ( r ≥ 2) symmetric matrix arguments have been defined, extending the zonal polynomials, and applied in multivariate distribution theory. The usefulness of the polynomials has attracted the attention of econometricians, and some recent papers have applied the methods to distribution theory in econometrics (e.g., Hillier [14] and Phillips [22]).

A test of uniformity on shape spaces

A test of uniformity on the shape space Σmk is presented, together with modifications of the test statistic which bring its null distribution close to the large-sample asymptotic distribution. The asymptotic distribution under suitable local alternatives to uniformity is given. A family of distributions on Σmk, is proposed, which is suitable for modelling shapes given by landmarks which are almost collinear.

Multivariate Meixner classes of invariant distributions

Abstract We define multivariate Meixner classes of invariant distributions of random matrices as those whose generating functions for the associated orthogonal polynomials are of certain special integral or summation forms, generalizing the univariate Meixner classes of distributions which were first characterized by Meixner [21]. Characterization theorems and properties of these multivariate Meixner classes are established. The zonal polynomials, the extended invariant polynomials with matrix arguments, and their related results in multivariate distribution theory are utilized in the discussion.

FUNCTIONAL FORMS OF CHARACTERISTIC FUNCTIONS AND CHARACTERIZATIONS OF MULTIVARIATE DISTRIBUTIONS

During the Oxford Conference of the Econometric Society in 1936, Ragnar Frisch proposed a problem of characterization of distributions based on the property of linear regression of one linear function of random variables on the other. This problem has been solved, partially by Allen [1], and then completely by Rao [24,25], Fix [7], and Laha [13] relaxing the conditions imposed on the component random variables. The purpose of this paper is to solve the above mentioned problem for the multivariate case, characterizing multivariate distributions based on the multivariate linear regression of one linear function of not necessarily i.i.d. random vectors with matrix coefficients on the other. We make some mild assumptions concerning the component random vectors and the related constant matrices. It is shown that the property of multivariate linear regression yields a system of partial differential equations (p.d.e.s) satisfied by the characteristic functions of the component random vectors. A general solution of this system of p.d.e.s is given by certain functional forms. Special cases of the general solution give characterizations of the “multivariate generalized stable laws” and the multivariate semistable laws, and a method is presented to characterize the multivariate stable laws.

Procrustes analysis on some special manifolds

We pnwnt some ilicorciical results obtained by applying Procrustes methods to the statistical analysis on the two special manifolds, the Stiefel manifold Vk,m and the Grassmann manifold Gk,m−k or, equivalently, the manifold Pk,m−k of all m × m orthogonal projection matrices idempoteut of rank k. Procrustes representations of Vk,m and Pk,m−k by means of equivalence classes of matrices are considered, and Procrustes statistics and means are defined via the ordinary, weighted and generalized Procrustes methods. We discuss perturbation theory in Procrustes analysis on Vk,m and Pk,m−k. Finally, we give a brief discussion of embeddings of the Stiefel and Grassmann manifolds.

Concentrated matrix Langevin distributions

This paper concerns the matrix Langevin distributions, exponential-type distributions defined on the two manifolds of our interest, the Stiefel manifold Vk,m and the manifold Pk,m-k of m × m orthogonal projection matrices idempotent of rank k which is equivalent to the Grassmann manifold Gk,m-k. Asymptotic theorems are derived when the concentration parameters of the distributions are large. We investigate the asymptotic behavior of distributions of some (matrix) statistics constructed based on the sample mean matrices in connection with testing hypotheses of the orientation parameters, and obtain asymptotic results in the estimation of large concentration parameters and in the classification of the matrix Langevin distributions.

Generalized noncentral hermite and laguerre polynomials in multiple matrices

Abstract We are concerned with the generalized “noncentral” Hermite and Laguerre polynomials in multiple matrices. We define the generalized Hermite polynomials H(mn)k[r];o(Y[q]; A[r]) in q m × n rectangular matrix arguments Y1,…,Yq (=Y[q]) and r m×m constant matrices A1,…, Ar (=A[r]) (q ⩽ r), associated with the joint distribution of q independent m×n rectangular matrix-variate standard normal distributions, and present various properties of theirs. The discussion of the generalized Hermite polynomials H(m)k[r];o(X[q]; A[r]) in q m × m symmetric matrix arguments X1,…, Xq (=X[q]), associated with the joint distribution of q independent m × m symmetric matrix-variate normal distributions, introduced by Chikuse, is further facilitated. The generalized noncentral Hermite polynomials are defined as H(mn)k[r];o((Y − N)[q]; A[r]) and H(m)k[r];o((X − M) [q]; A[r]) for q m × n rectangular and m × m symmetric matrices, N[q] and M[q], respectively. Extending results of Chikuse and Davis to the noncentral case, we define and discuss the generalized noncentral Laguerre polynomials L u[q] k[r];o (X [q] ; A [r] ; Ω [q] ) in q m × m noncentrality matrices Ω 1 ,…,Ω q (= Ω [q] ) , associated with the joint distribution of q independent noncentral Wishart distributions. Many unsolved problems can now be resolved by introducing these noncentral polynomials in multiple matrices. Specifically, both the differential and integral versions of the generalized multivariate Rodrigues formulae are obtained for all of these polynomials; they are of great use in multivariate distribution theory. A simple example of applications is presented.

Density estimation on the spaces of symmetric and rectangular matrices

Abstract This paper develops the theory of density estimation on the space S m of all m × m symmetric matrices and on the space R m , p of all m × p rectangular matrices. We consider the two methods, kernel density estimation and density estimation by orthogonal series, and illustrate them with examples for the normal kernel density functions and for the Hermite orthonormal bases with symmetric and rectangular matrix arguments on S m and R m , p , respectively. The problem of estimating unknown joint density functions of multiple random matrices is also discussed.