Consider a set of data consisting of n observations of a response variable Y and of vector of p explanatory variables X = (X 1, X 2, …, X p ) ⊤. Their relationship is described by the linear regression model (see Linear Regression Models)
In terms of the observed data, the model is
The variables e 1, …, e n are unobservable model errors, which are assumed being independent and identically distributed random variables with a distribution function F and density f. The density is unknown, we only assume that it is symmetric around 0. The vector β = (β 1, β 2, …, β p ) ⊤ is an unknown parameter, and the problem of interest is to estimate β based on observations Y 1, …, Y n and x i = (x i1, …, x ip ) ⊤, i = 1, …, n.
Besides the classical least squaresestimator, there exists a big...
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References and Further Reading
Arthanari TS, Dodge Y (1981) Mathematical programming in statistics. Wily, Interscience Division, New York; (1993) Wiley Classic Library
Dodge Y (1984) Robust estimation of regression coefficient by minimizing a convex combination of least squares and least absolute deviations. Comp Stat Quart 1:139–153
Dodge Y, Jurečková J (1987) Adaptive combination of least squares and least absolute deviations estimators. In: Dodge Y (ed) Statistical data analysis based on L 1 – norm and related methods. North-Holland, Amsterdam, pp 275–284
Dodge Y, Jurečková J (1988) Adaptive combination of M-estimator and L 1 – estimator in the linear model. In: Dodge Y, Fedorov VV, Wynn HP (eds) Optimal design and analysis of experiments. North-Holland, Amsterdam, pp 167–176
Dodge Y, Jurečková J (1991) Flexible L-estimation in the linear model. Comp Stat Data Anal 12:211–220
Dodge Y, Jurečková J (1995) Estimation of quantile density function based on regression quantiles. Stat Probab Lett 23: 73–78
Dodge Y, Jurečková J (2000) Adaptive regression. Springer, New York. ISBN 0-387-98965-X
Dodge Y, Lindstrom FT (1981) An alternative to least squares estimations when dealing with contaminated data. Technical report No 79, Oregon State University, Corvallis
Dodge Y, Antoch J, Jurečková J (1991) Adaptive combination of least squares and least absolute deviation estimators. Comp State Data Anal 12:87–99
Donoho DL, Huber PJ (1983) The notion of breakdown point. In: Bickel PJ, Doksum KA, Hodges JL (eds) A festschrift for Erich Lehmann. Wadsworth, Belmont, California
Hampel FR (1968) Contributions to the theory of robust estimation. PhD Thesis, University of California, Berkely
Koenker R (2005) Quantile regression. Cambridge University Press, Cambridge. ISBN 0-521-84573-4
Taylor LD (1973) Estimation by minimizing the sum of absolute errors. In: Zarembka P (ed) Frontiers in econometrics. Academic, New York, pp 189–190
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Jurečková, J. (2011). Adaptive Linear Regression. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_105
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