Abstract
In exploratory data analysis and curve fitting in particular, it is often desirable to observe residual values obtained with different estimation criteria. The goal with most linear model curve-fitting procedures is to minimize, in some sense, the vector of residuals. Perhaps three of the most common estimation criteria require minimizing: the sum of the absolute residuals (least absolute value or L1 norm); the sum of the squared residuals (least squares or L2 norm); and the maximum residual (Chebychev or L∞ norm). This paper demonstrates that utilizing the least squares residuals to provide an advanced start for the least absolute value and Chebychev procedures results in a significant reduction in computational effort. Computational results are provided.
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References
G. Appa and C. Smith, “On L1 and Chebyshev estimation”,Mathematical Programming 5 (1973) 73–87.
R.D. Armstrong and E.L. Frome, “A comparison of two algorithms for absolute deviation curve fitting”,Journal of the American Statistical Association 71 (1976) 328–330.
R.D. Armstrong, E. Frome and D. Kung, “A revised simplex algorithm for the absolute deviation curve fitting problem”,Communications in Statistics B8 (1978) 175–190.
R.D. Armstrong and D. Kung, “Minimax estimates for a linear multiple regression problem”,Applied Statistics 28 (1979) 93–100.
I. Barrodale and C. Phillips, “Solution of an overdetermined system of linear equations in the Chebychev norm”,ACM Transactions on Mathematical Software 1 (1975) 264–270.
I. Barrodale and F.D.K. Roberts, “An improved algorithm for discrete L1 linear approximation”,SIAM Journal of Numerical Analysis 10 (1977) 839–848.
I. Barrodale and A. Young, “Algorithms for best L1 and L∞ linear approximation on a discrete set”,Numerical Mathematics 8 (1966) 295–306.
J.E. Gentle, “Least absolute values estimation: An introduction”,Communications in Statistics—Simulation and Computation B6 (1977) 313–328.
M. Hand and V.A. Sposito, “Using the least squares estimator in Chebychev estimation”,Communications in Statistics—Simulation and Computation B9 (1980) 43–49.
IMSL: International Mathematical and Statistical Library, 7500 Bellaire, Houston, Texas 77036 (1979).
G.F. McCormick and V.A. Sposito, “Using the L2-estimator in L1-estimation”,SIAM Journal of Numerical Analysis 13 (1976).
R.S. Scowen, “Algorithm 271 Quickersort”,Communications of the ACM 8 (1965) 669–670.
V.A. Sposito,Linear and nonlinear programming (Iowa State University Press, Ames, Iowa, 1975).
V.A. Sposito, “Minimizing the maximum absolute deviation”,SIGMAP Newsletter 20 (1976) 51–53.
V.A. Sposito and M.L. Hand, “Using an approximate L1 estimator”,Communications in Statistics—Simulation and Computation B6 (1977) 263–268.
K. Spyropoulos, E. Kiountouzis and A. Young, “Discrete approximations in the L1 norm”,The Computer Journal 16 (1973) 180–186.
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Sklar, M.G., Armstrong, R.D. Least absolute value and chebychev estimation utilizing least squares results. Mathematical Programming 24, 346–352 (1982). https://doi.org/10.1007/BF01585115
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DOI: https://doi.org/10.1007/BF01585115