Abstract
Due to the complicated mathematical and nonlinear nature of ridge regression estimator, Liu (Linear-Unified) estimator has been received much attention as a useful method to overcome the weakness of the least square estimator, in the presence of multicollinearity. In situations where in the linear model, errors are far away from normal or the data contain some outliers, the construction of Liu estimator can be revisited using a rank-based score test, in the line of robust regression. In this paper, we define the Liu-type rank-based and restricted Liu-type rank-based estimators when a sub-space restriction on the parameter of interest holds. Accordingly, some improved estimators are defined and their asymptotic distributional properties are investigated. The conditions of superiority of the proposed estimators for the biasing parameter are given. Some numerical computations support the findings of the paper.
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References
Ahmed SE (2014) Penalty, shrinkage and pretest strategies: variable selection and estimation. Springer, New York
Akdeniz F, Akdeniz Duran E (2010) Liu-type estimator in semiparametric regression models. J Stat Comput Simul 80(8):853–871
Akdeniz F, Ozturk F (2005) The distribution of stochastic shrinkage biasing parameters of the Liu type estimator. Appl Math Comput 163(1):29–38
Akdeniz Duran E, Akdeniz F (2012) Efficiency of the modified jackknifed Liu-type estimator. Stat Pap 53(2):265–280
Akdeniz Duran E, Akdeniz F, Hu H (2011) Efficiency of a Liu-type estimator in semiparametric regression models. J Comput Appl Math 235(5):1418–1428
Arashi M, Kibria BMG, Norouzirad M, Nadarajah S (2014) Improved preliminary test and Stein-rule Liu estimators for the ill-conditioned elliptical linear regression model. J Multi Anal 124:53–74
Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96(456):1348–1360
Frank I, Friedman J (1993) A statistical view of some chemometrics regression tools (with discussion). Technometrics 35:109–148
Hettmansperger TP, McKean JW (1998) Robust nonparametric statistical methods. Arnold, London
Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation in non-orthogonal problems. Technometrics 12(3):55–67
Huang J, Breheny P, Ma S, Zhang CH (2010) The MNET method for variable selection (Unpublished Technical Report)
Kibria BMG (2004) Performance of the shrinkage preliminary test ridge regression estimators based on the conflicting of W, LR and LM tests. J Stat Comput Simul 74(11):703–810
Kibria BMG (2012) On some Liu and ridge type estimators and their properties under the ill-conditioned gaussian linear regression model. J Stat Comput Simul 82(1):1–17
Liu K (1993) A new class of biased estimate in linear regression. Commun Stat Theor Meth 22(2):393–402
Johnson BA, Peng L (2008) Rank-based variable selection. J Nonparameter Stat 20(3):241–252
Malthouse EC (1999) Shrinkage estimation and direct marketing scoring model. J Interact Mark 13(4):10–23
Puri ML, Sen PK (1986) A note on asymptotic distribution free tests for sub hypotheses in multiple linear regression. Ann Stat 1:553–556
Roozbeh M, Arashi M (2013) Feasible ridge estimator in partially linear models. J Multi Anal 116:35–44
Saleh AKMdE (2006) Theory of preliminary test and Stein-type estimation with applications. Wiley, New York
Saleh AKMdE, Kibria BMG (2011) On some ridge regression estimators: a nonparametric approach. J. Nonparametric Stat 23(3):819–851
Saleh AKMdE, Shiraishi T (1989) On some R and M-estimation of regression parameters under restriction. J Jpn Stat Soc 19(2):129–137
Sengupta D, Jammalamadaka SR (2003) Linear models: an integrated approach. World Scientific Publishing Company, Singapore
Stein C (1956) Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc Third Berkeley Symp 1:197–206
Tabatabaey SMM, Saleh AKMdE, Kibria BMG (2004) Estimation strategies for parameters of the linear regression model with spherically symmetric distributions. J Stat Res 38(1):13–31
Tibshirani R (1996) Regression shrinkage and selection via the LASSO. J R Stat Soc Ser B Stat Methodol 58(1):267–288
Yüzbaşı B, Ahmed SE, Güngör M (2017) Improved penalty strategies in linear regression models. REVSTAT-Stat J 15(2):251–276
Yüzbaşı B, Asar Y, Şık MŞ, Demiralp A (2017) Improving estimations in quantile regression model with autoregressive errors. Therm Sci. https://doi.org/10.2298/TSCI170612275Y
Xu J, Yang H (2012) On the Stein-type Liu estimator and positive-rule Stein-type Liu estimator in multiple linear regression models. Commun Stat Theor Meth 41(5):791–808
Zhang CH (2010) Nearly unbiased variable selection under minimax concave penalty. Ann Stat 38(2):894–942
Zou H (2006) The adaptive LASSO and its oracle properties. J Am Stat Assoc 101(476):1418–1429
Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J R Stat Soc Ser B Stat Methodol 67(2):301–320
Acknowledgements
We would like to thank two anonymous referees for their valuable and constructive comments which significantly improved the presentation of the paper and led to put many details. First author Mohammad Arashi’s work is based on the research supported in part by the National Research Foundation of South Africa (Grant NO. 109214). Third author S. Ejaz Ahmed is supported by the Natural Sciences and the Engineering Research Council of Canada (NSERC).
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Arashi, M., Norouzirad, M., Ahmed, S.E. et al. Rank-based Liu regression. Comput Stat 33, 1525–1561 (2018). https://doi.org/10.1007/s00180-018-0809-8
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DOI: https://doi.org/10.1007/s00180-018-0809-8