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Andrews, D. F., and A. M. Herzberg (1985). Data. Springer-Velag, New York. Arthanari, T. S., and Y. Dodge (1981). Mathematical Programming in Statistics. Wiley, New York. Barrodale, I., and F. D. K. Roberts (1974). Algorithm 478: Solution of an overdetermined system of equations in the I)-norm. Communications of the Association for Computing Machinery, vol. 17, pp. 319-320. Bloomfield, P., and W. Steiger (1983). Least Absolute Deviations: Theory, Applications, and Algorithms. Birkhauser, Boston. Dielman, T., and R. Pfaffenberger (1982). LA V (least absolute value) estimation in linear regression: a review. In S. H. Zanakis and J. S. Rustagi (eds.), Optimization in Statistics. North-Holland, New York. Dielman, T., and R. Pfaffenberger (1990). Tests of linear hypotheses and LA V estimation: a Monte Carlo comparison. Communications in Statistics-Simulation and Computation, vol. 19, pp. 1179-1199. Dodge, Y. (cd.) (1987a). Statistical Data Analysis Based on the L)-Norm and Related Methods. North-Holland, New York. Dodge, Y. (ed.) (1987b). Special Issue on Statistical Data Analysis Based on the L) Norm and Related Methods. Computational Statistics & Data Analysis, vol. 5, pp. 237-450. Dodge, Y. (cd.) (1992). L)-Statistical Analysis and Related Methods. North-Holland, Amsterdam. Farebrother, R. W. (1988). Algorithm AS 238: A simple recursive procedure for the L) norm fitting of a straight line. Applied Statistics, vol. 37, pp. 457-465. Gentle, J. E., S. C. Narula, and V. A. Sposito (1987). Algorithms for unconstrained L) linear regression. In Y. Dodge (ed.) (1987a). Hill, R. W., and P. W. Holland (1977). Two robust alternatives to least squares regression. Journal of the American Statistical Association, vol. 72, pp. 828-833. Koenker, R., and G. Bassett (1982). Tests of linear hypotheses and I) estimation. Econometrica, vol. 50, pp. 1577-1583. Marazzi, A. (1987). Solving bounded influence regression problems with ROBSYS. In Y. Dodge (ed.) (1987a).
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McKean, J. W., and R. M. Schrader (1987). Least absolute errors analysis of variance. In Y. Dodge (ed.) (1987a). Sadovski, A. N. (1974). Algorithm AS 74: L)-norm fit of a straight line. Applied Statistics, vol. 23, pp. 244-248. Schrader, R. M., and J. W. McKean (1987). Small sample properties of least absolute errors analysis of variance. In Y. Dodge (ed.) (1987a). Stigler, S. M. (1986). The History of Statistics: The Measurement of Uncertainty Before 1900. Harvard University Press, Cambridge, MA. Whyte, L. L. (ed.) (1961). Roger Joseph Boscouich. Allen & Unwin, London.
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M-regression and LAD regression are part of what is called robust statistics. A statistical procedure is regarded as "robust" if it performs reasonably well even when the assumptions of the statistical model are not true. If we assume that our data follow a normal linear regression model, then least-squares estimates and test perform quite well, but they are not robust when the assumption of normality for the population of random errors is invalid. M-regression was specifically developed to be robust with respect to this assumption. Peter Huber introduced the idea of M-estimation in 1964.
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Data from a study on the shelf life of packaged foods are shown in Table 5.1. Moisture content is the primary factor that determines shelf life, that is, the length of time during which the cereal is acceptable for eating. The higher the moisture content, the soggier the cereal, and hence the lower its acceptability. A particular brand of dry cereal was stored on a shelf at 73 F and 50% relative humidity. At various times during storage, the moisture content of the cereal was measured. We can use these data to see how moisture content depends on the time on the shelf. Time is measured in days. First we make a plot of the data with x = days on the shelf and y = moisture content. This is shown in Figure 5.1. We see that a straight line 85
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