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The high-dimensional data arises in diverse fields of sciences, engineering and humanities. Variable selection plays an important role in dealing with high dimensional statistical modelling. In this article, we study the variable selection of quadratic approximation via thesmoothly clipped absolute deviation(SCAD) penalty with a diverging number of parameters. We provide a unified method to select variables and estimate parameters for various of high dimensional models. Under appropriate conditions and with a proper regularization parameter, we show that the estimator has consistency and sparsity, and the estimators of nonzero coefficients enjoy the asymptotic normality as they would have if the zero coefficients were known in advance. In addition, under some mild conditions, we can obtain the global solution of the penalized objective function with the SCAD penalty. Numerical studies and a real data analysis are carried out to confirm the performance of the proposed method. [ABSTRACT FROM PUBLISHER]

When outliers and/or heavy-tailed errors exist in linear models, theleast absolute deviation(LAD) regression is a robust alternative to the ordinary least squares regression. Existing variable-selection methods in linear models based on LAD regression either only consider the finite number of predictors or lack the oracle property associated with the estimator. In this article, we focus on the variable selection via LAD regression with a diverging number of parameters. The rate of convergence of the LAD estimator with thesmoothly clipped absolute deviation(SCAD) penalty function is established. Furthermore, we demonstrate that, under certain regularity conditions, the penalized estimator with a properly selected tuning parameter enjoys the oracle property. In addition, the rank correlation screening method originally proposed by Liet al.(2011) is applied to deal with ultrahigh dimensional data. Simulation studies are conducted for revealing the finite sample performance of the estimator. We further illustrate the proposed methodology by a real example. [ABSTRACT FROM PUBLISHER]

In the economics and biological gene expression study area where a large number of variables will be involved, even when the predictors are independent, as long as the dimension is high, the maximum sample correlation can be large. Variable selection is a fundamental method to deal with such models. The ridge regression performs well when the predictors are highly correlated and some nonconcave penalized thresholding estimators enjoy the nice oracle property. In order to provide a satisfactory solution to the collinearity problem, in this paper we report the combined-penalization (CP) mixed by the nonconcave penalty and ridge, with a diverging number of parameters. It is observed that the CP estimator with a diverging number of parameters can correctly select covariates with nonzero coefficients and can estimate parameters simultaneously in the presence of multicollinearity. Simulation studies and a real data example demonstrate the well performance of the proposed method. [ABSTRACT FROM AUTHOR]

In survival studies, current status data are frequently encountered when some individuals in a study are not successively observed. This paper considers the problem of simultaneous variable selection and parameter estimation in the high-dimensional continuous generalized linear model with current status data. We apply the penalized likelihood procedure with the smoothly clipped absolute deviation penalty to select significant variables and estimate the corresponding regression coefficients. With a proper choice of tuning parameters, the resulting estimator is shown to be a rootn/pn-consistent estimator under some mild conditions. In addition, we show that the resulting estimator has the same asymptotic distribution as the estimator obtained when the true model is known. The finite sample behavior of the proposed estimator is evaluated through simulation studies and a real example. [ABSTRACT FROM PUBLISHER]

This article studies variable selection and parameter estimation in the partially linear model when the number of covariates in the linear part increases to infinity. Using the bridge penalty method, we succeed in selecting the important covariates of the linear part. Under regularity conditions, we have shown that the bridge penalized estimator of the parametric part enjoys the oracle property. We also obtain the convergence rate of the estimator of the nonparametric part. Simulation studies show that the bridge estimator performs as well as the oracle estimator for the partially linear model. An application is analyzed to illustrate the bridge procedure. [ABSTRACT FROM AUTHOR]

In this article, a robust variable selection procedure based on the weighted composite quantile regression (WCQR) is proposed. Compared with the composite quantile regression (CQR), WCQR is robust to heavy‐tailed errors and outliers in the explanatory variables. For the choice of the weights in the WCQR, we employ a weighting scheme based on the principal component method. To select variables with grouping effect, we consider WCQR with SCAD‐L2 penalization. Furthermore, under some suitable assumptions, the theoretical properties, including the consistency and oracle property of the estimator, are established with a diverging number of parameters. In addition, we study the numerical performance of the proposed method in the case of ultrahigh‐dimensional data. Simulation studies and real examples are provided to demonstrate the superiority of our method over the CQR method when there are outliers in the explanatory variables and/or the random error is from a heavy‐tailed distribution. [ABSTRACT FROM AUTHOR]

In applications, predictors are naturally grouped with some variables in nonzero groups being irrelevant, simultaneously variable selection at both the group and within-group levels is more desirable. In addition, to achieve a robust estimation against outliers in both covariates and responses, combining the excellent properties of weighted least absolute deviation (WLAD) and least squares, we propose an adjusted WLAD (AWLAD) regression estimator with the adaptive group bridge penalty. Importantly, we demonstrate that the AWLAD estimator enjoys the oracle property when the number of parameters grows with the sample size. Simulation studies and a real data analysis indicate that the AWLAD has superior performance in the finite sample cases. [ABSTRACT FROM AUTHOR]