Model Selection for Vector Autoregressive Processes via Adaptive Lasso

Determination of the best subset is an important step in vector autoregressive (VAR) modeling. Traditional methods either conduct subset selection and parameter estimation separately or compute expensively. In this article, we propose a VAR model selection procedure using adaptive Lasso, for it is computational efficient and can select subset and estimate parameters simultaneously. By proper choice of tuning parameters, we can choose the correct subset and obtain the asymptotic normality of the non zero parameters. Simulation studies and real data analysis show that adaptive Lasso performs better than existing methods in VAR model fitting and prediction.     

Variable Selection for Semiparametric Partially Linear Covariate-Adjusted Regression Models

In this article, the partially linear covariate-adjusted regression models are considered, and the penalized least-squares procedure is proposed to simultaneously select variables and estimate the parametric components. The rate of convergence and the asymptotic normality of the resulting estimators are established under some regularization conditions. With the proper choices of the penalty functions and tuning parameters, it is shown that the proposed procedure can be as efficient as the oracle estimators. Some Monte Carlo simulation studies and a real data application are carried out to assess the finite sample performances for the proposed method.     

Variable Selection for Partially Linear Models with Randomly Censored Data

This article proposes a variable selection procedure for partially linear models with right-censored data via penalized least squares. We apply the SCAD penalty to select significant variables and estimate unknown parameters simultaneously. The sampling properties for the proposed procedure are investigated. The rate of convergence and the asymptotic normality of the proposed estimators are established. Furthermore, the SCAD-penalized estimators of the nonzero coefficients are shown to have the asymptotic oracle property. In addition, an iterative algorithm is proposed to find the solution of the penalized least squares. Simulation studies are conducted to examine the finite sample performance of the proposed method.     

Lasso with convex loss: Model selection consistency and estimation

Abstract

Variable selection is a fundamental challenge in statistical learning if one works with data sets containing huge amount of predictors. In this artical we consider procedures popular in model selection: Lasso and adaptive Lasso. Our goal is to investigate properties of estimators based on minimization of Lasso-type penalized empirical risk with a convex loss function, in particular nondifferentiable. We obtain theorems concerning rate of convergence in estimation, consistency in model selection and oracle properties for Lasso estimators if the number of predictors is fixed, i.e. it does not depend on the sample size. Moreover, we study properties of Lasso and adaptive Lasso estimators on simulated and real data sets.     

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.     

Variable Selection in Semiparametric Quantile Modeling for Longitudinal Data

We propose a penalized quantile regression for partially linear varying coefficient (VC) model with longitudinal data to select relevant non parametric and parametric components simultaneously. Selection consistency and oracle property are established. Furthermore, if linear part and VC part are unknown, we propose a new unified method, which can do three types of selections: separation of varying and constant effects, selection of relevant variables, and it can be carried out conveniently in one step. Consistency in the three types of selections and oracle property in estimation are established as well. Simulation studies and real data analysis also confirm our method.     

Group Variable Selection with Oracle Property by Weight-Fused Adaptive Elastic Net Model for Strongly Correlated Data

This paper is the generalization of weight-fused elastic net (Fu and Xu, 2012 Fu, G., Xu, Q. (2012). Grouping variable selection by weight fused elastic net for multi-collinear data. Communications in Statistics-Simulation and Computation 41(2):205–221.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), which performs group variable selection by combining weight-fused LASSO(wfLasso) and elastic net (Zou and Hastie, 2005 Zou, H., Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67(2):301–320.[Crossref], [Web of Science ®] , [Google Scholar]) penalties. In this study, the elastic net penalty is replaced by adaptive elastic net penalty (AdaEnet) (Zou and Zhang, 2009 Zou, H., Zhang, H. (2009). On the adaptive elastic-net with a diverging number of parameters. Annals of Statistics 37(4):1733–1751.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), and a new group variable selection algorithm with oracle property (Fan and Li, 2001 Fan, J., Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96(456):1348–1360.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]; Zou, 2006 Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association 101(476):1418–1429.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is obtained.     

The Adaptive Gril Estimator with a Diverging Number of Parameters

We consider the problem of variables selection and estimation in linear regression model in situations where the number of parameters diverges with the sample size. We propose the adaptive Generalized Ridge-Lasso (mboxAdaGril) which is an extension of the the adaptive Elastic Net. AdaGril incorporates information redundancy among correlated variables for model selection and estimation. It combines the strengths of the quadratic regularization and the adaptively weighted Lasso shrinkage. In this article, we highlight the grouped selection property for AdaCnet method (one type of AdaGril) in the equal correlation case. Under weak conditions, we establish the oracle property of AdaGril which ensures the optimal large performance when the dimension is high. Consequently, it achieves both goals of handling the problem of collinearity in high dimension and enjoys the oracle property. Moreover, we show that AdaGril estimator achieves a Sparsity Inequality, i.e., a bound in terms of the number of non-zero components of the “true” regression coefficient. This bound is obtained under a similar weak Restricted Eigenvalue (RE) condition used for Lasso. Simulations studies show that some particular cases of AdaGril outperform its competitors.     

Prediction Error Property of the Lasso Estimator and its Generalization

The lasso procedure is an estimator‐shrinkage and variable selection method. This paper shows that there always exists an interval of tuning parameter values such that the corresponding mean squared prediction error for the lasso estimator is smaller than for the ordinary least squares estimator. For an estimator satisfying some condition such as unbiasedness, the paper defines the corresponding generalized lasso estimator. Its mean squared prediction error is shown to be smaller than that of the estimator for values of the tuning parameter in some interval. This implies that all unbiased estimators are not admissible. Simulation results for five models support the theoretical results.     

Adaptive penalized quantile regression for high dimensional data

We propose a new adaptive L₁ penalized quantile regression estimator for high-dimensional sparse regression models with heterogeneous error sequences. We show that under weaker conditions compared with alternative procedures, the adaptive L₁ quantile regression selects the true underlying model with probability converging to one, and the unique estimates of nonzero coefficients it provides have the same asymptotic normal distribution as the quantile estimator which uses only the covariates with non-zero impact on the response. Thus, the adaptive L₁ quantile regression enjoys oracle properties. We propose a completely data driven choice of the penalty level _λn${\lambda }_n$, which ensures good performance of the adaptive L₁ quantile regression. Extensive Monte Carlo simulation studies have been conducted to demonstrate the finite sample performance of the proposed method.     

Robust Variable Selection in Linear Mixed Models

In this article, we develop a robust variable selection procedure jointly for fixed and random effects in linear mixed models for longitudinal data. We propose a penalized robust estimator for both the regression coefficients and the variance of random effects based on a re-parametrization of the linear mixed models. Under some regularity conditions, we show the oracle properties of the proposed robust variable selection method. Simulation study shows the robustness of the proposed method against outliers. In the end, the proposed methods is illustrated in the analysis of a real data set.     

Variable Selection for Naive Bayes Semisupervised Learning

This article deals with a semisupervised learning based on naive Bayes assumption. A univariate Gaussian mixture density is used for continuous input variables whereas a histogram type density is adopted for discrete input variables. The EM algorithm is used for the computation of maximum likelihood estimators of parameters in the model when we fix the number of mixing components for each continuous input variable. We carry out a model selection for choosing a parsimonious model among various fitted models based on an information criterion. A common density method is proposed for the selection of significant input variables. Simulated and real datasets are used to illustrate the performance of the proposed method.     

Input Variable Selection in Neural Network Models

One of the most important issues in using neural networks for the analysis of real-world problems is the input variable selection problem. This article connects input variable selection with multiple testing in the neural network regression models. In the proposed procedure, the number and the type of input neurons are selected by means of a testing scheme, based on appropriate measures of relevance of a given input variable to the model. In order to avoid the data snooping problem, family-wise error rate is controlled by using the StepM method proposed by Romano and Wolf (2005 Romano , J. P. , Wolf , M. ( 2005 ). Exact and approximate stepdown methods for multiple hypothesis testing . J. Amer. Statist. Assoc. 100 : 94 – 108 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The testing procedure is calibrated by using the subsampling, which is shown to deliver consistent results under weak assumptions on the data generating process and on the structure of the neural network model.     

New Robust Variable Selection Methods for Linear Regression Models

Motivated by an entropy inequality, we propose for the first time a penalized profile likelihood method for simultaneously selecting significant variables and estimating unknown coefficients in multiple linear regression models in this article. The new method is robust to outliers or errors with heavy tails and works well even for error with infinite variance. Our proposed approach outperforms the adaptive lasso in both theory and practice. It is observed from the simulation studies that (i) the new approach possesses higher probability of correctly selecting the exact model than the least absolute deviation lasso and the adaptively penalized composite quantile regression approach and (ii) exact model selection via our proposed approach is robust regardless of the error distribution. An application to a real dataset is also provided.     

Oracle inequalities for the Lasso in the additive hazards model with interval-censored data

This article studies the absolute penalized convex function estimator in sparse and high-dimensional additive hazards model. Under such model, we assume that the failure time data are interval-censored and the number of time-dependent covariates can be larger than the sample size. We establish oracle inequalities based on some natural extensions of the compatibility and cone invertibility factors of the Hessian matrix at the true parameters in the model. Some similar inequalities based on an extension of the restricted eigenvalue are also established. Under mild conditions, we prove that the compatibility and cone invertibility factors and the restricted eigenvalues are bounded from below by positive constants for time-dependent covariates.     

Corrected proof of the result of 'A prediction error property of the Lasso estimator and its generalization' by Huang (2003)

The Lasso achieves variance reduction and variable selection by solving an ?₁‐regularized least squares problem. Huang (2003) claims that ‘there always exists an interval of regularization parameter values such that the corresponding mean squared prediction error for the Lasso estimator is smaller than for the ordinary least square estimator’. This result is correct. However, its proof in Huang (2003) is not. This paper presents a corrected proof of the claim, which exposes and uses some interesting fundamental properties of the Lasso.     

A New Analysis Strategy for Designs With Complex Aliasing

Abstract

Nonregular designs are popular in planning industrial experiments for their run-size economy. These designs often produce partially aliased effects, where the effects of different factors cannot be completely separated from each other. In this article, we propose applying an adaptive lasso regression as an analytical tool for designs with complex aliasing. Its utility compared to traditional methods is demonstrated by analyzing real-life experimental data and simulation studies.     

Performance of Variable Selection Methods in Regression Using Variations of the Bayesian Information Criterion

The Bayesian information criterion (BIC) is widely used for variable selection. We focus on the regression setting for which variations of the BIC have been proposed. A version that includes the Fisher Information matrix of the predictor variables performed best in one published study. In this article, we extend the evaluation, introduce a performance measure involving how closely posterior probabilities are approximated, and conclude that the version that includes the Fisher Information often favors regression models having more predictors, depending on the scale and correlation structure of the predictor matrix. In the image analysis application that we describe, we therefore prefer the standard BIC approximation because of its relative simplicity and competitive performance at approximating the true posterior probabilities.     

Variable selection in robust semiparametric modeling for longitudinal data

This paper considers robust variable selection in semiparametric modeling for longitudinal data with an unspecified dependence structure. First, by basis spline approximation and using a general formulation to treat mean, median, quantile and robust mean regressions in one setting, we propose a weighted M-type regression estimator, which achieves robustness against outliers in both the response and covariates directions, and can accommodate heterogeneity, and the asymptotic properties are also established. Furthermore, a penalized weighted M-type estimator is proposed, which can do estimation and select relevant nonparametric and parametric components simultaneously, and robustly. Without any specification of error distribution and intra-subject dependence structure, the variable selection method works beautifully, including consistency in variable selection and oracle property in estimation. Simulation studies also confirm our method and theories.     

Exponential-Bound Property of Estimators and Variable Selection in Generalized Additive Models

In this article, utilizing a scale mixture of skew-normal distribution in which mixing random variable is assumed to follow a mixture model with varying weights for each observation, we introduce a generalization of skew-normal linear regression model with the aim to provide resistant results. This model, which also includes the skew-slash distribution in a particular case, allows us to accommodate and detect outlying observations under the skew-normal linear regression model. Inferences about the model are carried out through the empirical Bayes approach. The conditions for propriety of the posterior and for existence of posterior moments are given under the standard noninformative priors for regression and scale parameters as well as proper prior for skewness parameter. Then, for Bayesian inference, a Markov chain Monte Carlo method is described. Since posterior results depend on the prior hyperparameters, we estimate them adopting the empirical Bayes method as well as using a Monte Carlo EM algorithm. Furthermore, to identify possible outliers, we also apply the Bayes factor obtained through the generalized Savage-Dickey density ratio. Examining the proposed approach on simulated instance and real data, it is found to provide not only satisfactory parameter estimates rather allow identifying outliers favorably.