A reformulation of the criteria for the independence of quadratic functions in normal variables

The Craig-Sakamoto theorem establishes a sufficient and necessary condition for the independence of two quadratic forms in normal variates, fascinating many statisticians and mathematicians, who continuously seek for simple and better proofs of the theorem and its extensions. In this article, we present a simple proof of a unified theorem on the independence of linear and quadratic functions in general normal variates.     

The Impact of Measurement Error on Principal Component Analysis

We investigate the effect of measurement error on principal component analysis in the high‐dimensional setting. The effects of random, additive errors are characterized by the expectation and variance of the changes in the eigenvalues and eigenvectors. The results show that the impact of uncorrelated measurement error on the principal component scores is mainly in terms of increased variability and not bias. In practice, the error‐induced increase in variability is small compared with the original variability for the components corresponding to the largest eigenvalues. This suggests that the impact will be negligible when these component scores are used in classification and regression or for visualizing data. However, the measurement error will contribute to a large variability in component loadings, relative to the loading values, such that interpretation based on the loadings can be difficult. The results are illustrated by simulating additive Gaussian measurement error in microarray expression data from cancer tumours and control tissues.     

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.     

All Invariant Moments of the Wishart Distribution

Abstract.  In this paper, we compute moments of a Wishart matrix variate U of the form E ( Q ( U )) where Q ( u ) is a polynomial with respect to the entries of the symmetric matrix u , invariant in the sense that it depends only on the eigenvalues of the matrix u . This gives us in particular the expected value of any power of the Wishart matrix U or its inverse U ^{− 1}. For our proofs, we do not rely on traditional combinatorial methods but rather on the interplay between two bases of the space of invariant polynomials in U . This means that all moments can be obtained through the multiplication of three matrices with known entries. Practically, the moments are obtained by computer with an extremely simple Maple program.     

Eigenvalue Ratio Test for the Number of Factors

This paper proposes two new estimators for determining the number of factors (r) in static approximate factor models. We exploit the well‐known fact that the r largest eigenvalues of the variance matrix of N response variables grow unboundedly as N increases, while the other eigenvalues remain bounded. The new estimators are obtained simply by maximizing the ratio of two adjacent eigenvalues. Our simulation results provide promising evidence for the two estimators.     

On the intrinsic force of convergence to stability∗

Observed populations differ greatly in the speed with which they approach the stable form, but what determines rates of convergence is not fully understood. The present paper shows that the force of convergence does not approach a fixed value, but oscillates indefinitely around an “intrinsic”; level. That level, h?, is determined by the square of the ratio of the 2 largest eigenvalues of the Leslie matrix. The value of h? can be closely approximated by a simple function that changes directly with the square of the coefficient of variation and inversely with the mean of the stable net maternity function. Population entropy, another measure of dispersion relative to the mean, is also highly correlated with h?.     

Branching Random Walks With Several Sources*

A continuous-time branching random walk on multidimensional lattices with a finite number of branching sources of three types leads to explicit conditions for the exponential growth of the total number of particles. These conditions are expressed in terms of the spectral characteristics of the operator describing the mean number of particles both at an arbitrary point and on the entire lattice.     

Direct methods for generating extreme characteristic roots of certain random matrices

Computer generation of extreme characteristic roots of random matrices is considered. The usual approach in Monte-Carlo applications is to randomly generate the matrix and then compute desired characteristic roots. There are, however, theoretical results about the distribution of individual characteristic roots which might be used as a basis for computing algorithms. This alternative approach is considered for the Wishart and Beta matrices.     

Continuity and Analysis of Sequences of Principal Components

Double arrays of n rows and p columns can be regarded as n drawings from some p-dimensional population. A sequence of such arrays is considered. Principal component analysis for each array forms sequences of sample principal components and eigenvalues. The continuity of these sequences, in the sense of convergence with probability one and convergence in probability, is investigated, that appears to be informative for pattern study and prediction of principal components. Various features of paths of sequences of population principal components are highlighted through an example.     

Estimating the number of factors to include in a high-dimensional multivariate bilinear model

We present two new statistics for estimating the number of factors underlying in a multivariate system. One of the two new methods, the original NUMFACT, has been used in high profile environmental studies. The two new methods are first explained from a geometrical viewpoint. We then present an algebraic development and asymptotic cutoff points. Next we present a simulation study that shows that for skewed data the new methods are typically superior to traditional methods and for normally distributed data the new methods are competitive to the best of the traditional methods. We finally show how the methods compare by using two environmental data sets.