Least squares and least absolute deviation procedures in approximately linear models

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., |b_i| ≤ r_i, r_i 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 |b_i| ≤ r_i. 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 r_i on b_i. This establishes another robustness property of the least absolute deviation criterion.