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The first five sections describeMultiple regression multiple regressionMultiple regression with least squaresLeast squares, including methods for outlierOutlier detection. Section 3.6 introduces three distinct approaches to robust regression: (i) soft trimming or downweighting, which extends the M-estimation of Chap.  2 to regression, principally S-estimation (Sect. 3.8); (ii) hard trimming, Least Trimmed SquaresLeast Trimmed Squares (LTS) (LTS, Sect. 3.11) in which a specified proportion of the observations is trimmed; and (iii) adaptive hard trimming, the Forward SearchForward search (FSFS, Sect.  4.1 ). This monitoring method is explored more thoroughly in Chap.  4 . Unlike the FSFS, the methods in (i) and (ii) provide a single robust analysis under chosen specified conditions. That for LTSLeast Trimmed Squares (LTS) depends on the chosen trimming proportion, which should, hopefully, trim all outliersOutlier and fit the model to all the uncontaminated data. For the downweighting methods, the severity of downweighting is determined by the choice of tuning constantsTuning constants to give desired robustnessRobustness properties. The calculations are described in Sect. 3.9. The algorithm for S-estimation is in Sect. 3.10 with that for LTSLeast Trimmed Squares (LTS) in Sect. 3.11.2. Further developments of M-estimation (MM- and \(\tau \) -estimators) are described in Sects. 3.12.1 and 3.12.2. Reweighted LTSLeast Trimmed Squares (LTS) estimatorsLeast trimmed squares estimator are introduced in Sect. 3.12.3. Section 3.13 concludes the chapter with comparisons of traditional robust data analyses for a single specified target of robustnessRobustness.

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Robust Estimators in Multiple Regression

  • 章节

  • 出版模式：

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  • 发表日期：

    2025年

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  • Anthony C. Atkinson,
  • Marco Riani,
  • Aldo Corbellini,
  • Domenico Perrotta,
  • Valentin Todorov

摘要

The first five sections describeMultiple regression multiple regressionMultiple regression with least squaresLeast squares, including methods for outlierOutlier detection. Section 3.6 introduces three distinct approaches to robust regression: (i) soft trimming or downweighting, which extends the M-estimation of Chap.  2 to regression, principally S-estimation (Sect. 3.8); (ii) hard trimming, Least Trimmed SquaresLeast Trimmed Squares (LTS) (LTS, Sect. 3.11) in which a specified proportion of the observations is trimmed; and (iii) adaptive hard trimming, the Forward SearchForward search (FSFS, Sect.  4.1 ). This monitoring method is explored more thoroughly in Chap.  4 . Unlike the FSFS, the methods in (i) and (ii) provide a single robust analysis under chosen specified conditions. That for LTSLeast Trimmed Squares (LTS) depends on the chosen trimming proportion, which should, hopefully, trim all outliersOutlier and fit the model to all the uncontaminated data. For the downweighting methods, the severity of downweighting is determined by the choice of tuning constantsTuning constants to give desired robustnessRobustness properties. The calculations are described in Sect. 3.9. The algorithm for S-estimation is in Sect. 3.10 with that for LTSLeast Trimmed Squares (LTS) in Sect. 3.11.2. Further developments of M-estimation (MM- and \(\tau \) -estimators) are described in Sects. 3.12.1 and 3.12.2. Reweighted LTSLeast Trimmed Squares (LTS) estimatorsLeast trimmed squares estimator are introduced in Sect. 3.12.3. Section 3.13 concludes the chapter with comparisons of traditional robust data analyses for a single specified target of robustnessRobustness.