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Robust linear regression using smooth adaptive estimators. (English) Zbl 0902.62078

Summary: We propose a robust simple linear regression method, namely the smooth adaptive line (SAL), which divides the data set into three equal groups on the basis of the ordered values of the explanatory variable. The estimators of the slope and intercept are obtained by using the smooth adaptive estimators (SA) of the three groups. The estimators are compared with the least squares (LS) estimators and two other three-group estimators, the resistant line (RL) method and the Bartlett’s line (BL) method. A Monte Carlo study is used to study their biases and relative efficiencies for the cases with and without outliers under either normality or non-normality assumption. When there is no outlier or one outlier or small outliers, SAL dominates RL for distributions with tails lighter than \(t_3\). Also, SAL dominates BL except for small sample size, say \(n = 10\). To compare SAL with LS, it is known that under normality assumption and no outlier, LS is the best. However, when there are outliers, SAL dominates LS when the outliers are in the \(x\)-direction or there are large outliers in the \(y\)-direction.

MSC:

62J05 Linear regression; mixed models
62F35 Robustness and adaptive procedures (parametric inference)
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[1] DOI: 10.2307/3001936 · doi:10.2307/3001936
[2] Brown G. W., Proccedings of the Second BerkcIey Symposium on Mathematical Statistles and Probability pp 159– (1951)
[3] DOI: 10.1080/03610929408831236 · Zbl 0825.62067 · doi:10.1080/03610929408831236
[4] DOI: 10.2307/2283768 · doi:10.2307/2283768
[5] DOI: 10.1080/03610928408828779 · Zbl 0552.62019 · doi:10.1080/03610928408828779
[6] Lo S., Robust Regression Using Smooth Adaptive Estimators (1996)
[7] Johnstone I. M., Proceedings of the Statistical Computing Section pp 218– (1982)
[8] DOI: 10.2307/2288572 · Zbl 0594.62077 · doi:10.2307/2288572
[9] Mood A. M., Introduction to the Theory of Statistics (1950) · Zbl 0039.13901
[10] Tukey J. W., Exploratory Data Analysis, Limited Preliminary Edition (1970)
[11] Vcllcman P. G., Applications, Basics, Computing of Exploratory Data Analysis (1981)
[12] Wald A., Annals of Mathematical Statistics 11 pp 282– (1940)
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