
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. Statistical Methods and Applications 08/2013; 22(3). DOI:10.1007/s102600130231x · 0.66 Impact Factor

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 (pseudounbiased) estimators that are symmetrically distributed about the true parameter values when the disturbances are symmetrically distributed about zero. L1Norm and L∞Norm Estimation, 01/2013: pages 3136;

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. L1Norm and L∞Norm Estimation, 01/2013: pages 4356;

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. L1Norm and L∞Norm Estimation, 01/2013: pages 1119;

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 onedimensional 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 twodimensional 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. L1Norm and L∞Norm Estimation, 01/2013: pages 59;

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 nonuniqueness of the solution to the \(L_1\)norm and \(L_{\infty }\)norm procedures. 01/2013: pages 2129;

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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):453456. DOI:10.1016/j.csda.2005.10.005 · 1.40 Impact Factor

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ABSTRACT: We establish that there are a total of 48 distinct ordered sets of three 4×4 (skewsymmetric) 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/S00243795(02)005359 · 0.94 Impact Factor

Richard William Farebrother ·
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ABSTRACT: This paper is concerned with the fitting of nonlinear 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 nonlinear function Journal of the Royal Statistical Society Series D (The Statistician) 01/2002; 47(1):137  147. DOI:10.1111/14679884.00119

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 timevarying allocation of votes between three political parties and with the timeinvariant 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):47780. DOI:10.1111/14679957.00259 · 0.26 Impact Factor

Richard William Farebrother ·
Technometrics 08/2000; 42(3). DOI:10.2307/1271125 · 1.81 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):121126. DOI:10.1016/S00243795(97)100064 · 0.94 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):60310. DOI:10.1111/14679957.00164 · 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(1264):312. DOI:10.1016/S00243795(96)003989 · 0.94 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 L1norm 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 reexamine some of the conclusions reached by GrattanGuinness (1970), Franksen (1985) and GrattanGuinness (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):455466. DOI:10.1016/S01679473(96)000710 · 1.40 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; 237238:205224. DOI:10.1016/00243795(95)006001 · 0.94 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 qdimensional hyperplane. This new criterion includes Rousseeuw's criteria as special cases. We also outline the corresponding criteria for fitting two or more qdimensional hyperplanes. Computational Statistics & Data Analysis 02/1995; 19(1):5358. DOI:10.1016/01679473(93)E00386 · 1.40 Impact Factor

Richard William Farebrother ·
SIAM Review 09/1994; 36(3). DOI:10.1137/1036121 · 2.91 Impact Factor

R. W. Farebrother ·
Applied Statistics 01/1990; 39(2). DOI:10.2307/2347778 · 1.49 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):719727. DOI:10.1080/03610918908812787 · 0.33 Impact Factor