Richard William Farebrother

Matrix representation of quaternions

Abstract We establish that there are a total of 48 distinct ordered sets of three 4×4 (skew-symmetric) signed permutation matrices which will serve as the basis of an algebra of quaternions.

Extending the relationship between ridge regression and continuum regression

Abstract Recently, a close relationship has been established between ridge regression (RR) and a special case of continuum regression (CR). Attention was restricted to the usual positive range of values for the ridge parameter. This restriction identifies the trajectory lying between ordinary least squares (OLS) and partial least squares (PLS) regressions, leaving the trajectory between PLS and principal component regression (PCR) untouched. In this note we demonstrate that the relationship between CR and RR can be extended to the full range of methods, OLS ↔ PLS ↔ PCR, identified by the CR technique. For this purpose one has to admit a nonstandard variant of the RR technique in which the ridge parameter becomes negative.

The historical development of the linear minimax absolute residual estimation procedure 1786–1960

Abstract This paper is concerned with the historical development of a traditional procedure for determining appropriate values for the parameters defining a linear relationship. This traditional procedure is variously known as the minimax absolute residual, Chebyshev, or L∞-norm procedure. Besides being of considerable interest in its own right as one of the earliest objective methods for estimating the parameters of such relationships, this procedure is also closely related to Rousseeuws least median of squared residuals and to the least sum of absolute residuals or L1-norm procedures. The minimax absolute residual procedure was first proposed by Laplace in 1786 and developed over the next 40 years by de Prony, Cauchy, Fourier, and Laplace himself. More recent contributions to this traditional literature include those of de la Vallee Poussin and Stiefel. Nowadays, the minimax absolute residual procedure is usually implemented as the solution of a primal or dual linear programming problem. It therefore comes as no surprise to discover that some of the more prominent features of such problems, including early variants of the simplex algorithm are to be found in these contributions. In this paper we re-examine some of the conclusions reached by Grattan-Guinness (1970), Franksen (1985) and Grattan-Guinness (1994) and suggest several amendments to their findings. In particular, we establish the nature of de Pronys geometrical fitting procedure and trace the origins of Fouriers prototype of the simplex algorithm.

On hyperplanes of closest fit

Abstract In this paper we generalise Rousseeuws least median squared residual and minimum volume ellipsoid criteria and obtain a suitable criterion for fitting a q -dimensional hyperplane. This new criterion includes Rousseeuws criteria as special cases. We also outline the corresponding criteria for fitting two or more q -dimensional hyperplanes.

A MEMOIR ON THE LIFE OF HAROLD THAYER DAVIS (1892^1974)*

In this paper we give brief details of the life of Harold Thayer Davis (1892–1974) and outline his contributions to econometrics in its early years.

SOME EARLY STATISTICAL CONTRIBUTIONS TO THE THEORY AND PRACTICE OF LINEAR ALGEBRA

Abstract We outline the history of some of the concepts and techniques of linear algebra which are intimately connected with the development of the method of least squares and related fitting procedures. Our study concentrates on contributions made during the early years of the nineteenth century, but it is not entirely restricted to this period.

Gauss, Carl Friedrich (1777–1855)

Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Germany) and died in Gottingen (Germany) on 23 February 1855. Gauss contributed to all areas of science known in his day, notably to astronomy, physics, and geodesy, but this article concentrates on his contributions to statistics and numerical methods. In 1805, Legendre published an algebraic procedure for the linear fitting problem that he named the method of least squares. Gausss first (1809) statistical derivation of this procedure was based on a combination of the normal distribution with a Bayesian variant of the maximum likelihood optimality criterion. Later, Gauss (1823–28) gave a very full account of an alternative decision-theoretic approach, which employed the mean squared error of the estimator as optimality criterion. In particular, he identified the least squares estimator as the linear unbiased estimator of the parameters with minimum mean squared error. Gauss developed the Gaussian elimination technique for solving determinate systems of linear equations, and explained how nonlinear relationships may be fitted by means of an iterative variant of the method of least squares. He also developed a second iterative (relaxation) procedure related to the Gauss–Seidel procedure.

A class of statistical estimators related to principal components

Abstract In this paper we investigate the algebraic relationships between some of the more familiar estimation and testing procedures employed in multivariate econometrics and the principal components and continuum regression techniques of multivariate statistics.

Short Communication: A linear programming procedure based on de la Vallée Poussin's minimax estimation procedure

It is shown that de la Vallee Poussins 1911 procedure for the solution of linear minimax estimation problems can be adjusted to solve a class of linear programming problems. A general procedure of this type should have been accessible in the 1910s, but the historical record shows that no such procedure was developed before the work of Kantorovich, Koopmans, and Dantzig in the 1940s.

On The Geometrical Representation of Allocations and Matrix Games

We describe the geometrical representation of allocations of quantities of goods, votes or probabilities between two or more persons, parties or strategies. We are particularly concerned with the representation of the time-varying allocation of votes between three political parties and with the time-invariant allocation of probabilities between the three strategies available to one of the participants in some matrix games. Copyright 2001 by Blackwell Publishers Ltd and The Victoria University of Manchester