Robust ridge M-estimators with pretest and Stein-rule shrinkage for an intercept term
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Robust ridge M‑estimators with pretest and Stein‑rule shrinkage for an intercept term Jia‑Han Shih1 · Ting‑Yu Lin2 · Masayuki Jimichi3 · Takeshi Emura4,5 Received: 26 September 2019 / Accepted: 5 September 2020 © Japanese Federation of Statistical Science Associations 2020
Abstract If the data contain both multicollinearity and outliers, the ridge M-estimator is the preferred estimator to the usual least square estimator (Silvapulle, Aust J Stat 33:319–333, 1991). Many other estimators, such as the pretest ridge M-estimator and Stein-rule shrinkage ridge M-estimator, have been developed on the basis of the ridge M-estimator. However, all these existing estimators do not consider shrinkage estimation for the intercept term. Hence, there are some rooms for improving the existing estimators by improving the estimator for the intercept term. In this paper, we propose several new ridge M-estimators for regression coefficients and an intercept term by introducing pretest and Stein-rule shrinkage schemes. Our estimators are obtained by using the Jimichi-type ridge matrix that allows shrinkage operations to be applicable to both the intercept term and regression coefficients. We conduct Monte Carlo simulation studies to examine the performance of the proposed estimators. For demonstration, we analyze the corporate finance data from the Nikkei Economic Electronic Databank System in Japan, and the gene expression data from Japanese ovarian cancer patients. Keywords Multicollinearity · Outlier · Ridge estimator · Robust estimator · Shrinkage estimator
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s4208 1-020-00089-6) contains supplementary material, which is available to authorized users. * Takeshi Emura [email protected] 1
Institute of Statistical Science, Academia Sinica, Taipei, Taiwan
2
Graduate Institute of Statistics, National Central University, Taoyuan, Taiwan
3
School of Business Administration, Kwansei Gakuin University, Nishinomiya, Japan
4
Department of Information Management, Chang Gung University, Taoyuan, Taiwan
5
Division of Hematology‑Oncology, Chang Gung Memorial Hospital at Linkou, No.259, Wenhua 1st Rd., Guishan Dist., Taoyuan 33302, Taiwan
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Vol.:(0123456789)
Japanese Journal of Statistics and Data Science
1 Introduction In a linear regression model, high correlations between explanatory variables are called multicollinearity. When multicollinearity exists, the ordinary least square estimator (LSE) has a large sampling variance. Ridge regression is an alternative approach proposed by Hoerl and Kennard (1970), which becomes one of the most popular methods to resolve multicollinearity problems. Ridge regression aims to reduce the large variance by shrinking the LSE toward zero. If the data contain outliers, the robust M-estimator (Huber 1981) is preferred to the LSE. If the data suffer both multicollinearity and outliers, Silvapulle (1991) suggested combining the ridge estimator with the M-estimator. Norouzirad and Arashi (2019)
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