TY - JOUR
T1 - Penalized weighted least absolute deviation regression
AU - Gao, Xiaoli
AU - Feng, Yang
N1 - Funding Information:
∗The authors would like to thank the Editor, the AE and two referees for the insightful comments which led to substantial improvement over an earlier version. †Partially supported by Simons Foundation #359337. ‡Partially supported by NSF CAREER grant DMS-1554804. §Corresponding author. E-mail address: [email protected].
PY - 2018
Y1 - 2018
N2 - In a linear model where the data is contaminated or the random error is heavy-tailed, least absolute deviation (LAD) regression has been widely used as an alternative approach to least squares (LS) regression. However, it is well known that LAD regression is not robust to outliers in the explanatory variables. When the data includes some leverage points, LAD regression may perform even worse than LS regression. In this manuscript, we propose to improve LAD regression in a penalized weighted least absolute deviation (PWLAD) framework. The main idea is to associate each observation with a weight reflecting the degree of outlying and leverage effect and obtain both the weight and coefficient vector estimation simultaneously and adaptively. The proposed PWLAD is able to provide regression coefficients estimate with strong robustness, and perform outlier detection at the same time, even when the random error does not have finite variances. We provide sufficient conditions under which PWLAD is able to identify true outliers consistently. The performance of the proposed estimator is demonstrated via extensive simulation studies and real examples.
AB - In a linear model where the data is contaminated or the random error is heavy-tailed, least absolute deviation (LAD) regression has been widely used as an alternative approach to least squares (LS) regression. However, it is well known that LAD regression is not robust to outliers in the explanatory variables. When the data includes some leverage points, LAD regression may perform even worse than LS regression. In this manuscript, we propose to improve LAD regression in a penalized weighted least absolute deviation (PWLAD) framework. The main idea is to associate each observation with a weight reflecting the degree of outlying and leverage effect and obtain both the weight and coefficient vector estimation simultaneously and adaptively. The proposed PWLAD is able to provide regression coefficients estimate with strong robustness, and perform outlier detection at the same time, even when the random error does not have finite variances. We provide sufficient conditions under which PWLAD is able to identify true outliers consistently. The performance of the proposed estimator is demonstrated via extensive simulation studies and real examples.
KW - Lasso
KW - Leverage points
KW - Outlier detection
KW - Robust regression
KW - Weighted least absolute deviation
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U2 - 10.4310/SII.2018.v11.n1.a7
DO - 10.4310/SII.2018.v11.n1.a7
M3 - Article
AN - SCOPUS:85039972021
SN - 1938-7989
VL - 11
SP - 79
EP - 89
JO - Statistics and its Interface
JF - Statistics and its Interface
IS - 1
ER -