George P. H. Styan

The Equality of the Ordinary Least Squares Estimator and the Best Linear Unbiased Estimator

Abstract It is well known that the ordinary least squares estimator of Xβ in the general linear model E y = Xβ, cov y = σ2 V, can be the best linear unbiased estimator even if V is not a multiple of the identity matrix. This article presents, in a historical perspective, the development of the several conditions for the ordinary least squares estimator to be best linear unbiased. Various characterizations of these conditions, using generalized inverses and orthogonal projectors, along with several examples, are also given. In addition, a complete set of references is provided.

Bounds for eigenvalues using traces

Several new inequalities are obtained for the modulus, the real part, and the imaginary part of a linear combination of the ordered eigenvalues of a square complex matrix. Included are bounds for the condition number, the spread, and the spectral radius. These inequalities involve the trace of a matrix and the trace of its square. Necessary and sufficient conditions for equality are given for each inequality.

Numerical computations for univariate linear models

We consider the usual univariate linear model In Part One of this paper X has full column rank. Numerically stable and efficient computational procedures are developed for the least squares estimation of y and the error sum of squares. We employ an orthogonal triangular decomposition of X using Householder Transformations. A lower bomd for the condition number of X is immediately obtained from this decomposition. Similar computational procedures are presented for the usual F-test of the general linear hypothesis L′γ=0; L′γ=m is also considered for m≠0. Updating techniques are given for adding to or removing from (X,y) a row, a set of rows or a column. In Part Two, X has less than full rank. Least squares estimates are obtained using generalized inverses. The function L′γ is estimable whenever it admits an unbiased estimator linear in y. We show how to computationally verify esthabiiity of L′γ and the equivalent testability of L′γ=0.

Rank equalities for idempotent and involutary matrices

Abstract We establish several rank equalities for idempotent and involutary matrices. In particular, we obtain new formulas for the rank of the difference, the sum, the product and the commutator of idempotent or involutary matrices. Extensions to scalar-potent matrices are also included. Our matrices are complex and are not necessarily Hermitian.

Computation of the stationary distribution of a markov chain

Eight algorithms are considered for the computation of the stationary distribution l´ of a finite Markov chain with associated probability transition matrix P. The recommended algorithm is based on solving l´(I—P+eu)=u, where e is the column vector of ones and u´ is a row vector satisfying u´e ≠0.An error analysis is presented for any such u including the choices u= ejP and u=e´j where ej is the jth row of the identity matrix. Computationalcomparisons between five of the algorithms are made based on twenty 8 x 8, twenty 20 x 20, and twenty 40 x 40 transition matrices. The matrix (I—P+eu)−1 is shown to be a non-singular generalized inverse of I—P when the unit root of P is simple and ue ≠ 0. A simple closed form expression is obtained for the Moore-Penrose inverse of I—P whenI—P has nullity one

Matrix Tricks for Linear Statistical Models

In teaching linear statistical models to first-year graduate students or to final-year undergraduate students there is no way to proceed smoothly without matrices and related concepts of linear algebra; their use is really essential. Our experience is that making some particular matrix tricks very familiar to students can substantially increase their insight into linear statistical models (and also multivariate statistical analysis). In matrix algebra, there are handy, sometimes even very simple “tricks” which simplify and clarify the treatment of a problem—both for the student and for the professor. Of course, the concept of a trick is not uniquely defined—by a trick we simply mean here a useful important handy result. In this book we collect together our Top Twenty favourite matrix tricks for linear statistical models.

Two matrix-based proofs that the linear estimator Gy is the best linear unbiased estimator

Abstract We offer two matrix-based proofs for the well-known result that the two conditions GX=X and GVQ=0 are necessary and sufficient for Gy to be the traditional best linear unbiased estimator (BLUE) of Xβ in the Gauss–Markov linear model {y,Xβ,V}, where y is an observable random vector with expectation vector E (y)=Xβ and dispersion matrix D (y)=V ; the matrix Q here is an arbitrary but fixed matrix whose range (column space) coincides with the null space of the transpose of X.

Rank Additivity and Matrix Polynomials.

Abstract : Various results are given concerning the logical relation between the rank additivity condition of matrices and polynomial equations satisfied by these matrices, generalizing earlier results on idempotent, tripotent, and r- potent matrices.

More Bounds for Elgenvalues Using Traces

Abstract Let the n × n complex matrix A have complex eigenvalues λ 1 ,λ 2 ,…λ n . Upper and lower bounds for Σ(Reλ i ) 2 are obtained, extending similar bounds for Σ|λ i | 2 obtained by Eberlein (1965), Henrici (1962), and Kress, de Vries, and Wegmann (1974). These bounds involve the traces of A ∗ A , B 2 , C 2 , and D 2 , where B = 1 2 ( A + A ∗ ) , C = 1 2 ( A − A ∗ ) /i , and D = AA ∗ − A ∗ A , and strengthen some of the results in our earlier paper “Bounds for eigenvalues using traces” in Linear Algebra and Appl. [12].

On a separation theorem for generalized eigenvalues and a problem in the analysis of sample surveys

Abstract We obtain usable bounds for the asymptotic percentage points of chi-squared tests of fit for log-linear models fitted to contingency tables estimated from survey data, by applying some new separation inequalities for the generalized eigenvalues of a matrix X′AX with respect to a matrix X′BX , when both the matrices A and B are nonnegative definite. We also present some historical remarks on the Poincare separation theorem for eigenvalues from which our new inequalities are shown to follow.