TY - JOUR

T1 - Least squares and least absolute deviation procedures in approximately linear models

AU - Mathew, Thomas

AU - Nordström, Kenneth

N1 - Funding Information:
* Research supported by Grant AFOSR 89-0237 and by a Special Research Initiative Support award from the Designated Research Initiative Fund, University of Maryland Baltimore County. * * On leave from University of Helsinki, Finland. Research supported by grants of the Deutsche Forschungsgemeinschaft and the Academy of Finland.
Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1993/1/27

Y1 - 1993/1/27

N2 - The Approximately Linear Model, introduced by Sacks and Ylvisaker (1978, The Annals of Statistics), represents deviations from the ideal linear model y = Xβ + e, by considering y = b + Xβ + e, where b is an unknown bias vector whose components are bounded in absolute value, i.e., |bi| ≤ ri, ri being a known nonnegative number. We propose to estimate β by minimizing the maximum of a weighted sum of squared deviations, or the sum of absolute deviations, where the maximum is computed subject to |bi| ≤ ri. In the former case the criterion to be minimized turns out to be a linear combination of the least squares and least absolute deviation criteria for the ideal linear model. The estimate of β obtained by the latter approach (i.e., by minimizing the maximum of a weighted sum of absolute deviations) turns out to be independent of the assumed bound ri on bi. This establishes another robustness property of the least absolute deviation criterion.

AB - The Approximately Linear Model, introduced by Sacks and Ylvisaker (1978, The Annals of Statistics), represents deviations from the ideal linear model y = Xβ + e, by considering y = b + Xβ + e, where b is an unknown bias vector whose components are bounded in absolute value, i.e., |bi| ≤ ri, ri being a known nonnegative number. We propose to estimate β by minimizing the maximum of a weighted sum of squared deviations, or the sum of absolute deviations, where the maximum is computed subject to |bi| ≤ ri. In the former case the criterion to be minimized turns out to be a linear combination of the least squares and least absolute deviation criteria for the ideal linear model. The estimate of β obtained by the latter approach (i.e., by minimizing the maximum of a weighted sum of absolute deviations) turns out to be independent of the assumed bound ri on bi. This establishes another robustness property of the least absolute deviation criterion.

KW - Approximately linear model

KW - least absolute deviation

KW - least squares

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U2 - 10.1016/0167-7152(93)90160-K

DO - 10.1016/0167-7152(93)90160-K

M3 - Article

AN - SCOPUS:38249005509

VL - 16

SP - 153

EP - 158

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

IS - 2

ER -