Robb J. Muirhead

Developments in Eigenvalue Estimation

The area of decision-theoretic multiparameter estimation in multivariate statistics has been one of intense activity and wide interest over the past few years. Many classical procedures revolve around the eigen structures of random and parameter matrices. Invariance and other considerations tend to focus a great deal of attention on the eigenvalues. The purpose of this paper is to review some of the work relating to eigenvalue estimation.

Inference in canonical correlation analysis

The asymptotic behavior, for large sample size, is given for the distribution of the canonical correlation coefficients. The result is used to examine the Bartlett-Lawley test that the residual population canonical correlation coefficients are zero. A marginal likelihood function for the population coefficients is obtained and the maximum marginal likelihood estimates are shown to provide a bias correction.

Estimating a Particular Function of the Multiple Correlation Coefficient

Abstract Let R denote the sample multiple correlation coefficient formed from a sample from a normal distribution with population multiple correlation coefficient . This article considers the problem of estimating the parameter using quadratic loss. The best unbiased estimate of θ is a linear function of Y = R 2/(1 − R 2); it is shown that this is dominated by other linear estimates and that these, in turn, are dominated by nonlinear estimates. A Monte Carlo study indicates that such estimates perform much better than the best unbiased estimate in terms of mean squared error.

A Note on Some Wishart Expectations

SummaryIn a recent paper Sharma and Krishnamoorthy (1984) used a complicated decisiontheoretic argument to derive an identity involving expectations taken with respect to the Wishart distributionWm(n, I). A more general result, proved using an elementary moment generating function argument, and some applications, are given in this paper.

Estimating functions of canonical correlation coefficients

Let ρ21,…,ρ2p be the squares of the population canonical correlation coefficients from a normal distribution. This paper is concerned with the estimation of the parameters δ1,…,δp, where δi = ρ2i(1 − ρ2i), i = 1,…,p, in a decision theoretic way. The approach taken is to estimate a parameter matrix Δ whose eigenvalues are δ1,…,δp, given a random matrix F whose eigenvalues have the same distribution as r2i(1 − r2i), i = 1,…,p, where r1,…,rp are the sample canonical correlation coefficients.

Polynomial estimation of eigenvalues

In estimating the eigenvalues of the covariance matrix of a multivariate normal population, the usual estimates are the eigenvalues of the sample covariance matrix. It is well known that these estimates are biased. This paper investigates obtaining improved eigenvalue estimates through improved estimates of the characteristic polynomial, which is a function of the sample eigenvalues. A numerical study investigates the improvements evaluated under both a square error and an entropy loss function.

On some distribution problems in Manova and discriminant analysis

Asymptotic expansions, valid for large error degrees of freedom, are given for the multivariate noncentral F distribution and for the distribution of latent roots in MANOVA and discriminant analysis. The asymptotic results are expressed in terms of elementary functions which are easy to compute and the results of some numerical work are included. The Bartlett test of the null hypothesis that some of the noncentrality parameters in discriminant analysis are zero is also briefly discussed.