Richard William Farebrother

The University of Manchester, Manchester, England, United Kingdom

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Publications (29)27.99 Total impact

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    Richard William Farebrother
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    ABSTRACT: It is shown that de la Vallée Poussin's 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.
    Computational Statistics & Data Analysis 02/2006; 51(2):453-456. DOI:10.1016/j.csda.2005.10.005 · 1.15 Impact Factor
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    Richard William Farebrother, Jürgen Groß, Sven-Oliver Troschke
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    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.
    Linear Algebra and its Applications 03/2003; 362:251–255. DOI:10.1016/S0024-3795(02)00535-9 · 0.98 Impact Factor
  • Richard William Farebrother
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    ABSTRACT: This paper is concerned with the fitting of non-linear relationships such as the logistic and Gompertz functions. Although Gauss had proposed the now standard Gauss–Newton procedure for this purpose in 1809, and it was strongly championed by Schultz in 1930, this procedure did not come into common use until modern computing equipment was introduced in the 1960s. In its stead a variety of virtually arbitrary procedures were employed. These arbitrary procedures are still used when the practitioner requires preliminary estimates of the parameters of a given non-linear function
    Journal of the Royal Statistical Society Series D (The Statistician) 01/2002; 47(1):137 - 147. DOI:10.1111/1467-9884.00119
  • Richard William Farebrother
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    ABSTRACT: 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
    Manchester School 09/2001; 69(4):477-80. DOI:10.1111/1467-9957.00259 · 0.26 Impact Factor
  • Richard William Farebrother
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    ABSTRACT: This bock gives historical and mathematical descriptions of the fitting of linear relationships by the procedures of least absolute deviations and the minimax absolute deviations and by the method of least squares. The description of historical backgrounds for the corresponding periods and the lifes of some scientists are also given. Astronomical, geodetic and navigation problems were the first to require solutions of the fitting of linear relationships. First, practicioners determined the values of the q unknown parameter constants by selecting a set of q equations from the n avaiable. As an alternative to the method of selected points, Boscovich minimized the sum of the absolute errors subject to an adding-up constraint. Laplace’s discussion of the Boscovich procedure was an alternative to the minimax procedure. Legendre publicated in 1805 his description of the method of least squares. Gauss made significant contributions to the method of least absolute errors in 1809. In the same year he described a fitting procedure based on the principle of maximum probability and the normal law. In 1774 Laplace adopted an inverse (or Bayesian) approach to the fitting problem. Gauss derived in 1809 the normal law from the Principle of the Arithmetic Mean and the Principle of Maximum Probability. Adrain developed the normal law independently of Gauss. In 1818 Laplace developed an asymptotic theory for the least absolute errors procedure and proposed an alternative derivation of the method of least squares. In 1823 Gauss gave a very full treatment of the subject. He studied also the computational problems. Gauss solved (1828) the problem of fitting a linear model subject to a set of linear constraints. Practical variants of his procedures were developed by Doolitle and Cholesky. Donkin (1844) identified the concept of potential energy as a suitable analogy for the statistical concept underlying the method of least squares. Bravais (1846) adopted a geometrical approach. Laplace (1816) developed a sequential least squares method. Thiele (1897, 1903) gave a method based on the orthogonal-triangular decomposition. Edgeworth (1888, 1923) developed the methods for determining two constants in such a way that they minimize the sum of the absolute deviations.
    Technometrics 08/2000; 42(3). DOI:10.2307/1271125 · 1.79 Impact Factor
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    Richard William Farebrother
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    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.
    Linear Algebra and its Applications 03/1999; 289(1):121-126. DOI:10.1016/S0024-3795(97)10006-4 · 0.98 Impact Factor
  • Richard William Farebrother
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    ABSTRACT: 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.
    Manchester School 02/1999; 67(4):603-10. DOI:10.1111/1467-9957.00164 · 0.26 Impact Factor
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    Richard William Farebrother
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    ABSTRACT: We briefly outline the origins of formal matrix theory in the 1870s and discuss Aitken's role in the dissemination of matrix methods in the 1940s with particular reference to the subject area of statistics and economics.
    Linear Algebra and its Applications 10/1997; 264(264):3-12. DOI:10.1016/S0024-3795(96)00398-9 · 0.98 Impact Factor
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    Richard William Farebrother
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    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 Rousseeuw's 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 Vallée 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 Prony's geometrical fitting procedure and trace the origins of Fourier's prototype of the simplex algorithm.
    Computational Statistics & Data Analysis 06/1997; 24(4):455-466. DOI:10.1016/S0167-9473(96)00071-0 · 1.15 Impact Factor
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    Richard William Farebrother
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    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.
    Linear Algebra and its Applications 04/1996; 237-238:205-224. DOI:10.1016/0024-3795(95)00600-1 · 0.98 Impact Factor
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    Richard William Farebrother
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    ABSTRACT: In this paper we generalise Rousseeuw's least median squared residual and minimum volume ellipsoid criteria and obtain a suitable criterion for fitting a q-dimensional hyperplane. This new criterion includes Rousseeuw's criteria as special cases. We also outline the corresponding criteria for fitting two or more q-dimensional hyperplanes.
    Computational Statistics & Data Analysis 02/1995; 19(1):53-58. DOI:10.1016/0167-9473(93)E0038-6 · 1.15 Impact Factor
  • Richard William Farebrother
    SIAM Review 09/1994; 36(3). DOI:10.1137/1036121 · 4.79 Impact Factor
  • R. W. Farebrother
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    ABSTRACT: This paper describes an ISO Pascal algorithm.
    Applied Statistics 01/1990; DOI:10.2307/2347778 · 1.42 Impact Factor
  • R. W. Farebrother
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    ABSTRACT: In this paper we will show that certain features of the type of duality recently discussed by Souvaine and Steele (1987) and Owen and Shiau (1988) may be traced in the earlier work of Laplace (1786), Newcomb (1873a, b) and Edgeworth (1888).
    Communication in Statistics- Simulation and Computation 01/1989; 18(2):719-727. DOI:10.1080/03610918908812787 · 0.29 Impact Factor
  • R W. Farebrother
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    ABSTRACT: In this paper we generalize Hawkins, Bradu and Kass's concept of “elemental predicted residuals” to models in which the augmented regressor matrix X = [X′ R′]' has full column rank where X is the regressor matrix and RB=r are a set of linear constraints on the parameters of the model. We will show that the two-way classification model takes this form and will identify Bradu and Hawkins's tetrads as elemental predicted residuals in this context.
    Communication in Statistics- Theory and Methods 01/1988; 17(1):79-85. DOI:10.1080/03610928808829611 · 0.28 Impact Factor
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    R.W. Farebrother
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    ABSTRACT: In a recent paper on the theory of Euclidean distance matrices, Gower derived an inequality which characterizes such matrices and challenged his readers to provide a direct proof of this result. The present paper represents the author's response to that challenge.
    Linear Algebra and its Applications 10/1987; 95:11-16. DOI:10.1016/0024-3795(87)90024-3 · 0.98 Impact Factor
  • R.W. Farebrother
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    ABSTRACT: In this paper we examine the statistical foundations of the five tests for heteroscedasticity considered by Griffiths and Surekha (1986). We identify the precise specification of the alternative hypothesis as an important determinant of the power of the tests.
    Journal of Econometrics 02/1987; 36(3-36):359-368. DOI:10.1016/0304-4076(87)90007-8 · 1.53 Impact Factor
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    R. W. Farebrother
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    ABSTRACT: In this paper we outline Black and Newing's theory of committee decisions and examine its relationship to Bowley and Edgeworth's solutions of the estimation problem.
    Computational Statistics & Data Analysis 02/1987; 5(4):437-442. DOI:10.1016/0167-9473(87)90066-1 · 1.15 Impact Factor
  • R.W. Farebrother
    Journal of the Royal Statistical Society Series C Applied Statistics 01/1987; 36(3). DOI:10.2307/2347804 · 1.42 Impact Factor
  • R.W. Farebrother
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    ABSTRACT: In this paper we will show that the expressions obtained by Hillier (1986) for the null distribution of the likelihood ratio test of zero restrictions on nonnegative regression coefficients in the standard linear model are the same as those reported by Farebrother (1986).
    Communication in Statistics- Theory and Methods 01/1987; 16(7):2003-2005. DOI:10.1080/03610928708829486 · 0.28 Impact Factor