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In this article, we consider the penalized LAD regression which deals with variable selection and estimation simultaneously for single-index models. The proposed estimator is robust and efficient with respect to heavy tailed errors or outliers in the response. Then, we introduce a practical algorithm where the unknown link function is estimated by local linear smoothing combined with LAD regression and the parametric index is estimated through linear LAD regression. Furthermore, we show that, under certain appropriate conditions, the penalized LAD estimator has desired large sample properties including-consistency, and oracle property. Simulation studies and a real data application are carried to illustrate the performances of the proposed method for finite-sample cases. Finally, based on the idea of sure independence screening procedure and the distance correlation (DC-SIS) proposed by Li et al. (2012), a robust two-step estimator is introduced to deal with ultra-high dimensional data. [ABSTRACT FROM AUTHOR]

In this paper, we consider the shrinkage and penalty estimation procedures in the linear regression model with autoregressive errors of orderpwhen it is conjectured that some of the regression parameters are inactive. We develop the statistical properties of the shrinkage estimation method including asymptotic distributional biases and risks. We show that the shrinkage estimators have a significantly higher relative efficiency than the classical estimator. Furthermore, we consider the two penalty estimators: least absolute shrinkage and selection operator (LASSO) and adaptive LASSO estimators, and numerically compare their relative performance with that of the shrinkage estimators. A Monte Carlo simulation experiment is conducted for different combinations of inactive predictors and the performance of each estimator is evaluated in terms of the simulated mean-squared error. This study shows that the shrinkage estimators are comparable to the penalty estimators when the number of inactive predictors in the model is relatively large. The shrinkage and penalty methods are applied to a real data set to illustrate the usefulness of the procedures in practice. [ABSTRACT FROM PUBLISHER]

Variable selection in multiple linear regression models is considered. It is shown that for the special case of orthogonal predictor variables, an adaptive pre-test-type procedure proposed by Venter and Steel [Simultaneous selection and estimation for the some zeros family of normal models, J. Statist. Comput. Simul. 45 (1993), pp. 129–146] is almost equivalent to least angle regression, proposed by Efron et al. [Least angle regression, Ann. Stat. 32 (2004), pp. 407–499]. A new adaptive pre-test-type procedure is proposed, which extends the procedure of Venter and Steel to the general non-orthogonal case in a multiple linear regression analysis. This new procedure is based on a likelihood ratio test where the critical value is determined data-dependently. A practical illustration and results from a simulation study are presented. [ABSTRACT FROM AUTHOR]