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The group Lasso is a penalized regression method, used in regression problems where the covariates are partitioned into groups to promote sparsity at the group level [27]. Quantile group Lasso, a natural extension of quantile Lasso [25], is a good alternative when the data has group information and has many outliers and/or heavy tails. How to discover important features that are correlated with interest of outcomes and immune to outliers has been paid much attention. In many applications, however, we may also want to keep the flexibility of selecting variables within a group. In this paper, we develop a sparse group variable selection based on quantile methods which select important covariates at both the group level and within the group level, which penalizes the empirical check loss function by the sum of square root group-wiseL1-norm penalties. The oracle properties are established where the number of parameters diverges. We also apply our new method to varying coefficient model with categorial effect modifiers. Simulations and real data example show that the newly proposed method has robust and superior performance. [ABSTRACT FROM PUBLISHER]

In this paper, we propose a new full iteration estimation method for quantile regression (QR) of the single-index model (SIM). The asymptotic properties of the proposed estimator are derived. Furthermore, we propose a variable selection procedure for the QR of SIM by combining the estimation method with the adaptive LASSO penalized method to get sparse estimation of the index parameter. The oracle properties of the variable selection method are established. Simulations with various non-normal errors are conducted to demonstrate the finite sample performance of the estimation method and the variable selection procedure. Furthermore, we illustrate the proposed method by analyzing a real data set. [ABSTRACT FROM PUBLISHER]

The semiparametric partially linear varying coefficient models (SPLVCM) are frequently used in statistical modelling, but most existing methods were built on either the least-square or likelihood-based methods, which are very sensitive to the outliers and their efficiency may be significantly reduced for heavy tail error distribution. This paper proposes a new efficient and robust estimation procedure for the SPLVCM based on modal regression. We establish the asymptotic normality of proposed estimators for both the parametric and nonparametric parts, and show that the estimators achieve the best convergence rate. Moreover, we develop a variable selection procedure to select significant parametric components for the SPLVCM and prove the method possessing the oracle property. We also discuss the bandwidth selection and propose an expectation–maximisation-type algorithm for the proposed estimation procedure. Some simulation results and real data analysis confirm that the newly proposed method works very competitively compared to other existing methods. [ABSTRACT FROM AUTHOR]