Richard William Farebrother

The University of Manchester, Manchester, England, United Kingdom

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Publications (37)29.21 Total impact

  • Richard William Farebrother
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    ABSTRACT: In this article we show that several leading natural scientists, statisticians and social scientists born between 1730 and 1930 are closely related by marriage, thereby forming what Annan (Studies in social history: a tribute to C. M. Trevelyan, Longmans, Green, London, pp 241–287, 1955) has named an Intellectual Aristocracy. We also establish that the first three individuals mentioned in our title had family connections with Italy.
    No preview · Article · Aug 2013 · Statistical Methods and Applications
  • Richard William Farebrother
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    ABSTRACT: Supposing that the disturbance terms in the standard linear statistical model are independent and follow a common Laplacian, Gaussian, or rectangular distribution, then the principle of maximum likelihood suggests that we should choose estimates of the slope parameters to minimise the \(L_t\)-norm of the residuals with \(t=1,\ t=2\) or \(t=\infty \) respectively. In this context, we outline the small sample and asymptotic theory relating to these maximum likelihood estimators and the related Likelihood Ratio, Lagrange Multiplier and Wald tests of linear restrictions on the parameters of the model. We also demonstrate that a simple modification of the standard linear programming implementation of the \(l_1\) -norm or \(L_{\infty }\)-norm fitting problem yields (pseudo-unbiased) estimators that are symmetrically distributed about the true parameter values when the disturbances are symmetrically distributed about zero.
    No preview · Chapter · Jan 2013
  • Richard William Farebrother
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    ABSTRACT: We describe a set of mechanical models that may be used to represent the various \(L_1\)-norm, \(L_2\)-norm and \(L_{\infty }\)-norm fitting procedures: the \(L_1\)-norm estimation problems may be represented by the positioning of a ring or a rigid rod under the influence of a frictionless system of strings and pulleys; the \(L_2\)-norm estimation problems may be represented by the positioning of a ring or a rigid rod under the influence of a frictionless system of stretched springs; and, by combining disparate aspects from these two mechanical models, we find that \( L_{\infty }\)-norm estimation problems may be represented by the positioning of a ring or a rigid rod under the influence of a system of strings and blocks. In the first two cases the optimal position of the ring or rigid rod is determined by a minimisation of the total potential energy of the system. In the third case we only have to determine the physical limitations imposed by the lengths of string attached to the ring or rod. Moreover, the mechanical model for the \(L_1\)-norm problem may be generalised to cover Oja’s bivariate median and the \(L_{\infty }\)-norm model may be generalised to cover Rousseeuw’s least median of squares problem.
    No preview · Chapter · Jan 2013
  • Richard William Farebrother
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    ABSTRACT: We continue our analysis by considering the fitting of a \(p\)-dimensional hyperplane to a set of point observations in \((p+1)\)-dimensional space. Again using the \(L_{t}\)-norm as optimality criterion with \(t=1, t=2\) or \(t=\infty \), we obtain the least absolute residuals, least squared residuals and minimax absolute residual procedures for the fitting of the hyperplane to this set of observations. Expressing the \(L_t\)-norm problem in matrix form, we establish that the weighted \(L_1\)-norm problem is intimately associated with a transformation of the weighted \(L_{\infty }\)-norm problem and vice versa. Then, examining the matrix representation of the \(L_1\)-norm and \(L_{\infty }\)-norm problems, we identify a particular vector as the Lagrange multipliers of these problems. Finally, we define the corresponding matrix expression in the \(L_2\)-norm case and identify it as the formulation of the least squares problem employed in continuum regression analysis.
    No preview · Chapter · Jan 2013
  • Richard William Farebrother
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    ABSTRACT: We begin our analysis by considering the fitting of a single point to a number of point observations in one-dimensional space. Using the \(L_{t}\)-norm as optimality criterion with \(t=1,\, t=2\) or \(t=\infty \), we obtain the median, mean and midrange of a set of observations respectively. Similarly, applying the same three optimality criteria in the two-dimensional case, we obtain the mediancentre or centre of population, the centroid or centre of gravity and the unnamed centre of the circle of smallest radius respectively. Moreover, if we omit some of the more extreme observations then we obtain truncated variants of these procedures. As noted in Chap. 7, the midrange and its generalisations may be associated with a set of more or less familiar geometrical instruments: The univariate midrange with a pair of callipers, the bivariate midrange with a pair of compasses and the minimax fitted line of Chap. 3 with a pair of parallel rules.
    No preview · Chapter · Jan 2013
  • Richard William Farebrother

    No preview · Chapter · Jan 2013
  • Richard William Farebrother
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    ABSTRACT: After sketching the graphical solution of the \(L_1\) -norm and \(L_{\infty }\)-norm fitting problems based on a plot of lines in parameter space, we survey the pertinent literature on modern standard and improved linear programming solutions to these problems. We investigate the possibility that a variant of the familiar simplex procedure could have been developed some thirty years before it actually appeared in the late 1940s. Finally, we survey a range of possible alternatives to using the \(L_t\)-norm procedure in the limit as \(t\) tends to \(1\) or \(\infty \) as possible practical solutions to the problem of non-uniqueness of the solution to the \(L_1\)-norm and \(L_{\infty }\)-norm procedures.
    No preview · Chapter · Jan 2013
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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.
    Preview · Article · Feb 2006 · Computational Statistics & Data Analysis
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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.
    Preview · Article · Mar 2003 · Linear Algebra and its Applications
  • 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
    No preview · Article · Jan 2002 · Journal of the Royal Statistical Society Series D (The Statistician)
  • 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
    No preview · Article · Sep 2001 · Manchester School
  • Richard William Farebrother

    No preview · Article · Aug 2000 · Technometrics
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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.
    Preview · Article · Mar 1999 · Linear Algebra and its Applications
  • 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.
    No preview · Article · Feb 1999 · Manchester School
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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.
    Preview · Article · Oct 1997 · Linear Algebra and its Applications
  • R.W. 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.
    No preview · Article · Jun 1997 · Computational Statistics & Data Analysis
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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.
    Preview · Article · Jun 1997 · Computational Statistics & Data Analysis
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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.
    Preview · Article · Apr 1996 · Linear Algebra and its Applications
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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.
    Preview · Article · Feb 1995 · Computational Statistics & Data Analysis
  • Richard William Farebrother

    No preview · Article · Sep 1994 · SIAM Review