## Abstract

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.

Original language | English |
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Pages (from-to) | 153-158 |

Number of pages | 6 |

Journal | Statistics and Probability Letters |

Volume | 16 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 27 1993 |

## Keywords

- Approximately linear model
- least absolute deviation
- least squares

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty