Quantile regression for robust estimation and variable selection in partially linear varying-coefficient models

In this paper, we develop a new estimation procedure based on quantile regression for semiparametric partially linear varying-coefficient models. The proposed estimation approach is empirically shown to be much more efficient than the popular least squares estimation method for non-normal error distributions, and almost not lose any efficiency for normal errors. Asymptotic normalities of the proposed estimators for both the parametric and nonparametric parts are established. To achieve sparsity when there exist irrelevant variables in the model, two variable selection procedures based on adaptive penalty are developed to select important parametric covariates as well as significant nonparametric functions. Moreover, both these two variable selection procedures are demonstrated to enjoy the oracle property under some regularity conditions. Some Monte Carlo simulations are conducted to assess the finite sample performance of the proposed estimators, and a real-data example is used to illustrate the application of the proposed methods.     

Bayesian variable selection in a finite mixture of linear mixed-effects models

Mixture of linear mixed-effects models has received considerable attention in longitudinal studies, including medical research, social science and economics. The inferential question of interest is often the identification of critical factors that affect the responses. We consider a Bayesian approach to select the important fixed and random effects in the finite mixture of linear mixed-effects models. To accomplish our goal, latent variables are introduced to facilitate the identification of influential fixed and random components and to classify the membership of observations in the longitudinal data. A spike-and-slab prior for the regression coefficients is adopted to sidestep the potential complications of highly collinear covariates and to handle large p and small n issues in the variable selection problems. Here we employ Markov chain Monte Carlo (MCMC) sampling techniques for posterior inferences and explore the performance of the proposed method in simulation studies, followed by an actual psychiatric data analysis concerning depressive disorder.     

Parametric simultaneous robust inferences for regression coefficient under generalized linear models

In this article, the parametric robust regression approaches are proposed for making inferences about regression parameters in the setting of generalized linear models (GLMs). The proposed methods are able to test hypotheses on the regression coefficients in the misspecified GLMs. More specifically, it is demonstrated that with large samples, the normal and gamma regression models can be properly adjusted to become asymptotically valid for inferences about regression parameters under model misspecification. These adjusted regression models can provide the correct type I and II error probabilities and the correct coverage probability for continuous data, as long as the true underlying distributions have finite second moments.     

Robust rank-based variable selection in double generalized linear models with diverging number of parameters under adaptive Lasso

We propose a robust rank-based estimation and variable selection in double generalized linear models when the number of parameters diverges with the sample size. The consistency of the variable selection procedure and asymptotic properties of the resulting estimators are established under appropriate selection of tuning parameters. Simulations are performed to assess the finite sample performance of the proposed estimation and variable selection procedure. In the presence of gross outliers, the proposed method is showing that the variable selection method works better. For practical application, a real data application is provided using nutritional epidemiology data, in which we explore the relationship between plasma beta-carotene levels and personal characteristics (e.g. age, gender, fat, etc.) as well as dietary factors (e.g. smoking status, intake of cholesterol, etc.).     

On some stepdown procedures with application to consistent variable selection in linear regression

The paper considers the problem of consistent variable selection with the use of stepdown procedures in the classical linear regression model and for the model with dependent errors. The stated results complete the results obtained by Bunea et al. [Consistent variable selection in high dimensional regression via multiple testing. J Stat Plann Inference. 2006;136(12):4349–4364].     

Variable selection for high-dimensional generalized linear models with the weighted elastic-net procedure

High-dimensional data arise frequently in modern applications such as biology, chemometrics, economics, neuroscience and other scientific fields. The common features of high-dimensional data are that many of predictors may not be significant, and there exists high correlation among predictors. Generalized linear models, as the generalization of linear models, also suffer from the collinearity problem. In this paper, combining the nonconvex penalty and ridge regression, we propose the weighted elastic-net to deal with the variable selection of generalized linear models on high dimension and give the theoretical properties of the proposed method with a diverging number of parameters. The finite sample behavior of the proposed method is illustrated with simulation studies and a real data example.     

Instrumental variable based variable selection for generalized linear models with endogenous covariates

We study the variable selection problem for a class of generalized linear models with endogenous covariates. Based on the instrumental variable adjustment technology and the smooth-threshold estimating equation (SEE) method, we propose an instrumental variable based variable selection procedure. The proposed variable selection method can attenuate the effect of endogeneity in covariates, and is easy for application in practice. Some theoretical results are also derived such as the consistency of the proposed variable selection procedure and the convergence rate of the resulting estimator. Further, some simulation studies and a real data analysis are conducted to evaluate the performance of the proposed method, and simulation results show that the proposed method is workable.     

WLAD-LASSO method for robust estimation and variable selection in partially linear models

This paper focuses on robust estimation and variable selection for partially linear models. We combine the weighted least absolute deviation (WLAD) regression with the adaptive least absolute shrinkage and selection operator (LASSO) to achieve simultaneous robust estimation and variable selection for partially linear models. Compared with the LAD-LASSO method, the WLAD-LASSO method will resist to the heavy-tailed errors and outliers in the parametric components. In addition, we estimate the unknown smooth function by a robust local linear regression. Under some regular conditions, the theoretical properties of the proposed estimators are established. We further examine finite-sample performance of the proposed procedure by simulation studies and a real data example.     

A simultaneous variable selection methodology for linear mixed models

Selecting an appropriate structure for a linear mixed model serves as an appealing problem in a number of applications such as in the modelling of longitudinal or clustered data. In this paper, we propose a variable selection procedure for simultaneously selecting and estimating the fixed and random effects. More specifically, a profile log-likelihood function, along with an adaptive penalty, is utilized for sparse selection. The Newton-Raphson optimization algorithm is performed to complete the parameter estimation. By jointly selecting the fixed and random effects, the proposed approach increases selection accuracy compared with two-stage procedures, and the usage of the profile log-likelihood can improve computational efficiency in one-stage procedures. We prove that the proposed procedure enjoys the model selection consistency. A simulation study and a real data application are conducted for demonstrating the effectiveness of the proposed method.     

Adaptive Lasso for generalized linear models with a diverging number of parameters

In this paper, we study the asymptotic properties of the adaptive Lasso estimators in high-dimensional generalized linear models. The consistency of the adaptive Lasso estimator is obtained. We show that, if a reasonable initial estimator is available, under appropriate conditions, the adaptive Lasso correctly selects covariates with non zero coefficients with probability converging to one, and that the estimators of non zero coefficients have the same asymptotic distribution they would have if the zero coefficients were known in advance. Thus, the adaptive Lasso has an Oracle property. The results are examined by some simulations and a real example.     

A computational framework for variable selection in multivariate regression

Stepwise variable selection procedures are computationally inexpensive methods for constructing useful regression models for a single dependent variable. At each step a variable is entered into or deleted from the current model, based on the criterion of minimizing the error sum of squares (SSE). When there is more than one dependent variable, the situation is more complex. In this article we propose variable selection criteria for multivariate regression which generalize the univariate SSE criterion. Specifically, we suggest minimizing some function of the estimated error covariance matrix: the trace, the determinant, or the largest eigenvalue. The computations associated with these criteria may be burdensome. We develop a computational framework based on the use of the SWEEP operator which greatly reduces these calculations for stepwise variable selection in multivariate regression.     

Structure identification and variable selection in geographically weighted regression models

Geographically weighted regression (GWR) is an important tool for exploring spatial non-stationarity of a regression relationship, in which whether a regression coefficient really varies over space is especially important in drawing valid conclusions on the spatial variation characteristics of the regression relationship. This paper proposes a so-called GWGlasso method for structure identification and variable selection in GWR models. This method penalizes the loss function of the local-linear estimation of the GWR model by the coefficients and their partial derivatives in the way of the adaptive group lasso and can simultaneously identify spatially varying coefficients, nonzero constant coefficients and zero coefficients. Simulation experiments are further conducted to assess the performance of the proposed method and the Dublin voter turnout data set is analysed to demonstrate its application.     

Adaptive elastic net-penalized quantile regression for variable selection

Abstract

There has been much attention on the high-dimensional linear regression models, which means the number of observations is much less than that of covariates. Considering the fact that the high dimensionality often induces the collinearity problem, in this article, we study the penalized quantile regression with the elastic net (EnetQR) that combines the strengths of the quadratic regularization and the lasso shrinkage. We investigate the weak oracle property of the EnetQR under mild conditions in the high dimensional setting. Moreover, we propose a two-step procedure, called adaptive elastic net quantile regression (AEnetQR), in which the weight vector in the second step is constructed from the EnetQR estimate in the first step. This two-step procedure is justified theoretically to possess the weak oracle property. The finite sample properties are performed through the Monte Carlo simulation and a real-data analysis.     

Variable selection for multivariate generalized linear models

Generalized linear models (GLMs) are widely studied to deal with complex response variables. For the analysis of categorical dependent variables with more than two response categories, multivariate GLMs are presented to build the relationship between this polytomous response and a set of regressors. Traditional variable selection approaches have been proposed for the multivariate GLM with a canonical link function when the number of parameters is fixed in the literature. However, in many model selection problems, the number of parameters may be large and grow with the sample size. In this paper, we present a new selection criterion to the model with a diverging number of parameters. Under suitable conditions, the criterion is shown to be model selection consistent. A simulation study and a real data analysis are conducted to support theoretical findings.     

Testing for varying zero-inflation and dispersion in generalized Poisson regression models

Homogeneity of dispersion parameters and zero-inflation parameters is a standard assumption in zero-inflated generalized Poisson regression (ZIGPR) models. However, this assumption may be not appropriate in some situations. This work develops a score test for varying dispersion and/or zero-inflation parameter in the ZIGPR models, and corresponding test statistics are obtained. Two numerical examples are given to illustrate our methodology, and the properties of score test statistics are investigated through Monte Carlo simulations.     

Marginal likelihood estimation from the Metropolis output: tips and tricks for efficient implementation in generalized linear latent variable models

The marginal likelihood can be notoriously difficult to compute, and particularly so in high-dimensional problems. Chib and Jeliazkov employed the local reversibility of the Metropolis–Hastings algorithm to construct an estimator in models where full conditional densities are not available analytically. The estimator is free of distributional assumptions and is directly linked to the simulation algorithm. However, it generally requires a sequence of reduced Markov chain Monte Carlo runs which makes the method computationally demanding especially in cases when the parameter space is large. In this article, we study the implementation of this estimator on latent variable models which embed independence of the responses to the observables given the latent variables (conditional or local independence). This property is employed in the construction of a multi-block Metropolis-within-Gibbs algorithm that allows to compute the estimator in a single run, regardless of the dimensionality of the parameter space. The counterpart one-block algorithm is also considered here, by pointing out the difference between the two approaches. The paper closes with the illustration of the estimator in simulated and real-life data sets.     

Variable selection in partially linear additive models for modal regression

Based on B-spline basis functions and smoothly clipped absolute deviation (SCAD) penalty, we present a new estimation and variable selection procedure based on modal regression for partially linear additive models. The outstanding merit of the new method is that it is robust against outliers or heavy-tail error distributions and performs no worse than the least-square-based estimation for normal error case. The main difference is that the standard quadratic loss is replaced by a kernel function depending on a bandwidth that can be automatically selected based on the observed data. With appropriate selection of the regularization parameters, the new method possesses the consistency in variable selection and oracle property in estimation. Finally, both simulation study and real data analysis are performed to examine the performance of our approach.