Takesi Hayakawa

The likelihood ratio criterion and the asymptotic expansion of its distribution

SummaryAsymptotic expansion of the distribution of the likelihood ratio criterion (LRC) for testing a composite hypothesis is derived under null hypothesis and a correction factor ρ which makes the term of order 1/n in the asymptotic expansion of the distribution of it vanish is obtained. The problem is extended to the case of a general composite hypothesis and of Pitmans local alternatives. The asymptotic distribution of LRC for a simple hypothesis is studied under a fixed alternative.

Asymptotic expansions of the distributions of some test statistics

A modified Wald statistic for testing simple hypothesis against fixed as well as local alternatives is proposed. The asymptotic expansions of the distributions of the proposed statistic as well as the Wald and Rao statistics under both the null and alternative hypotheses are obtained. The powers of these statistics are compared and its is shown that for special structures of parameters some statistics have same power in the sence of order\({1 \mathord{\left/ {\vphantom {1 {\sqrt n }}} \right. \kern-\nulldelimiterspace} {\sqrt n }}\). The results obtained are applied for testing the hypothesis about the covariance matrix of the multivariate normal distribution and it is shown that none of the tests based on the above statistics is uniformly superior.

On the distribution of the latent roots of a positive definite random symmetric matrix I

Many problems in multivariate analysis trr~tolve t~e distribution problems of the latent roots of positive definite random symmetric matrices. In particular, the distributions of the latent roots of a Wishart matrix and those of a multivariate quadratic form are very fundamental in the normal multivariate case. In this paper, we shall give the density functions of the following statistics composed of the latent roots of a non-central Wishart matrix and of a non-central multivariate quadratic form. ( i ) The latent roots of the determinantal equations det (~-2XX ~)=0 and det(X-2XAXP)=0 (sections 5.1 and 7.1), (ii) the maximum latent root (sections 5.2 and 7.2), (iii) the traces (sections 5.3 and 7.3). To treat the distribution problems of the latent roots of a non~ central Wishart matrix, we shall introduce a generalized Hermite polynomial with a matrix argument, discuss some properties of it and give its generating function (section 3). We shall also introduce a new function which is appropriate to discuss the distribution of a non-central multivariate quadratic form (section 6).

On the distribution of a quadratic form in a multivariate normal sample

SummaryThe distribution of a quadratic form in a normal sample plays a very important role in multivariate statistical analysis. In many cases, statistics are functions of a quadratic form or special types of it.In the univariate case, the distribution of a quadratic form was treated by many authors and was derived by using the Laguerre polynomials expansion or the Dirichlet series expansion, etc. [6], [7], [8]. In this paper, the distribution of a quadratic form in the multivariate case will be given in terms of zonal polynomials which were developed for multivariate analysis by A. T. James [3], [4] and A. G. Constantine [2]. Recently, the author’s attention was called to C. G. Khatri [9] which deals with the same problem also by using zonal polynomials. However, the present paper treats the problem from another point of view.The distribution of a quadratic form enables us to derive the distribution of a linear combination of several Wishart matrices. The distributions and probability functions of certain statistics of a quadratic form are given.

Normalizing and variance stabilizing transformations of multivariate statistics under an elliptical population

SummaryNormalizing and variance stabilizing transformations of a sample correlation, multiple correlation and canonical correlation coefficients are obtained under an elliptical population. It is shown that the Fishersz-transformation is efficient for these statistics. A normalizing transformation is also studied for a latent root of a sample covariance matrix in an elliptical sample.

On the distribution of the latent roots of a complex Wishart matrix (non-central case)

This paper considers the derivation of the probability density function of the latent roots of a non-central complex Wishart matrix. To treat this problem, we define the generalized Hermite polynomials of a complex matrix argument and give some properties of the generalized Hermite polynomials. By using the generating function of the generalized Hermite polynomials, we can obtain the exact probability density functions of the latent roots,. the maximum latent root and of trace of the latent roots of a non-centr~l complex Wishart matrix. The unitary invariant measure with total volume unity. The unitary invariant measure. The semi-unitary invariant measure with total volume ~nity. The zonal polynomial which corresponds the partition K of k into not more than m parts. exp(tr A).

On the distribution of the maximum latent root of a positive definite symmetric random matrix

In this paper, we consider the distribution of the maximum latent root of a certain positive definite symmetric random matrix. For this purpose, we give a useful transformation of a symmetric matrix and calculate its Jacobian. We also give some useful expansion formulas for zonal polynomials (A. T. James [3]).

On tests for the mean direction of the Langevin distribution

The asymptotic expansions of the distribution of a sum of independent random vectors with Langevin distribution are given. The power functions of the likelihood ratio criterion, Watson statistic, Rao statistic and the modified Wald statistic for testing the hypothesis of the mean direction are obtained asymptotically and a numerical comparison is made.

The asymptotic distributions of the statistics based on the complex Gaussian distribution

The asymptotic distributions of the statistics based on the complex Gaussian distribution. ABSTRACT This paper considers the asymptotic distributions of the statistics based on the complex Gaussian distribution. To treat this problem, we define the hermitian differential operator matrix and give the fundamental formulas of the zonal polynomials.

On testing hypothesis of covariance matrices under elliptical population

Abstract The power functions of the likehood ratio criteria and Rao statistics for testing hypotheses of the structures of covariance matrices of normal population are derived asymptotically and are compared numerically under an elliptical population.