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Least squares and least absolute deviation procedures in approximately linear models

Author

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  • Mathew, Thomas
  • Nordström, Kenneth

Abstract

The Approximately Linear Model, introduced by Sacks and Ylvisaker (1978, The Annals of Statistics), represents deviations from the ideal linear model y = X[beta] + e, by considering y = b + X[beta] + e, where b is an unknown bias vector whose components are bounded in absolute value, i.e., bi [less-than-or-equals, slant] ri, ri being a known nonnegative number. We propose to estimate [beta] 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 [less-than-or-equals, slant] 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 [beta] 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.

Suggested Citation

  • Mathew, Thomas & Nordström, Kenneth, 1993. "Least squares and least absolute deviation procedures in approximately linear models," Statistics & Probability Letters, Elsevier, vol. 16(2), pages 153-158, January.
    • Handle: RePEc:eee:stapro:v:16:y:1993:i:2:p:153-158
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