Variable selection approach for zero-inflated count data via adaptive lasso

This article proposes a variable selection approach for zero-inflated count data analysis based on the adaptive lasso technique. Two models including the zero-inflated Poisson and the zero-inflated negative binomial are investigated. An efficient algorithm is used to minimize the penalized log-likelihood function in an approximate manner. Both the generalized cross-validation and Bayesian information criterion procedures are employed to determine the optimal tuning parameter, and a consistent sandwich formula of standard errors for nonzero estimates is given based on local quadratic approximation. We evaluate the performance of the proposed adaptive lasso approach through extensive simulation studies, and apply it to analyze real-life data about doctor visits.     

A computationally fast alternative to cross-validation in penalized Gaussian graphical models

We study the problem of selecting a regularization parameter in penalized Gaussian graphical models. When the goal is to obtain a model with good predictive power, cross-validation is the gold standard. We present a new estimator of Kullback–Leibler loss in Gaussian Graphical models which provides a computationally fast alternative to cross-validation. The estimator is obtained by approximating leave-one-out-cross-validation. Our approach is demonstrated on simulated data sets for various types of graphs. The proposed formula exhibits superior performance, especially in the typical small sample size scenario, compared to other available alternatives to cross-validation, such as Akaike's information criterion and Generalized approximate cross-validation. We also show that the estimator can be used to improve the performance of the Bayesian information criterion when the sample size is small.     

Variable selection in linear measurement error models via penalized score functions

We propose variable selection procedures based on penalized score functions derived for linear measurement error models. To calibrate the selection procedures, we define new tuning parameter selectors based on the scores. Large-sample properties of these new tuning parameter selectors are established for the proposed procedures. These new methods are compared in simulations and a real-data application with competing methods where one ignores measurement error or uses the Bayesian information criterion to choose the tuning parameter.     

Simultaneous variable and factor selection via sparse group lasso in factor analysis

This paper considers variable and factor selection in factor analysis. We treat the factor loadings for each observable variable as a group, and introduce a weighted sparse group lasso penalty to the complete log-likelihood. The proposal simultaneously selects observable variables and latent factors of a factor analysis model in a data-driven fashion; it produces a more flexible and sparse factor loading structure than existing methods. For parameter estimation, we derive an expectation-maximization algorithm that optimizes the penalized log-likelihood. The tuning parameters of the procedure are selected by a likelihood cross-validation criterion that yields satisfactory results in various simulation settings. Simulation results reveal that the proposed method can better identify the possibly sparse structure of the true factor loading matrix with higher estimation accuracy than existing methods. A real data example is also presented to demonstrate its performance in practice.     

Nonlinear GCV and quasi-GCV for shrinkage models

The generalized cross-validation (GCV) method has been a popular technique for the selection of tuning parameters for smoothing and penalty, and has been a standard tool to select tuning parameters for shrinkage models in recent works. Its computational ease and robustness compared to the cross-validation method makes it competitive for model selection as well. It is well known that the GCV method performs well for linear estimators, which are linear functions of the response variable, such as ridge estimator. However, it may not perform well for nonlinear estimators since the GCV emphasizes linear characteristics by taking the trace of the projection matrix. This paper aims to explore the GCV for nonlinear estimators and to further extend the results to correlated data in longitudinal studies. We expect that the nonlinear GCV and quasi-GCV developed in this paper will provide similar tools for the selection of tuning parameters in linear penalty models and penalized GEE models.     

Variable selection via penalized minimum φ-divergence estimation in logistic regression

We propose penalized minimum φ-divergence estimator for parameter estimation and variable selection in logistic regression. Using an appropriate penalty function, we show that penalized φ-divergence estimator has oracle property. With probability tending to 1, penalized φ-divergence estimator identifies the true model and estimates nonzero coefficients as efficiently as if the sparsity of the true model was known in advance. The advantage of penalized φ-divergence estimator is that it produces estimates of nonzero parameters efficiently than penalized maximum likelihood estimator when sample size is small and is equivalent to it for large one. Numerical simulations confirm our findings.     

ON CROSS-VALIDATION FOR SMOOTHING SPLINES IN THE CASE OF DEPENDENT OBSERVATIONS

Cross-validation, as a popular tool for choosing a smoothing parameter, is generalized to the case of dependent observations. A general version of the ‘deletion theorem’ for representation and simplified calculation of cross-validatory criteria is given. Finally cross-validation is discussed in terms of penalized likelihoods as a method for model choice analogous to the Akaike information criterion.     

A regularized variable selection procedure in additive hazards model with stratified case-cohort design

Case-cohort designs are commonly used in large epidemiological studies to reduce the cost associated with covariate measurement. In many such studies the number of covariates is very large. An efficient variable selection method is needed for case-cohort studies where the covariates are only observed in a subset of the sample. Current literature on this topic has been focused on the proportional hazards model. However, in many studies the additive hazards model is preferred over the proportional hazards model either because the proportional hazards assumption is violated or the additive hazards model provides more relevent information to the research question. Motivated by one such study, the Atherosclerosis Risk in Communities (ARIC) study, we investigate the properties of a regularized variable selection procedure in stratified case-cohort design under an additive hazards model with a diverging number of parameters. We establish the consistency and asymptotic normality of the penalized estimator and prove its oracle property. Simulation studies are conducted to assess the finite sample performance of the proposed method with a modified cross-validation tuning parameter selection methods. We apply the variable selection procedure to the ARIC study to demonstrate its practical use.     

Shrinkage tuning parameter selection with a diverging number of parameters

Summary.  Contemporary statistical research frequently deals with problems involving a diverging number of parameters. For those problems, various shrinkage methods (e.g. the lasso and smoothly clipped absolute deviation) are found to be particularly useful for variable selection. Nevertheless, the desirable performances of those shrinkage methods heavily hinge on an appropriate selection of the tuning parameters. With a fixed predictor dimension, Wang and co-worker have demonstrated that the tuning parameters selected by a Bayesian information criterion type criterion can identify the true model consistently. In this work, similar results are further extended to the situation with a diverging number of parameters for both unpenalized and penalized estimators. Consequently, our theoretical results further enlarge not only the scope of applicabilityation criterion type criteria but also that of those shrinkage estimation methods.     

Tuning Parameter Selection in Cox Proportional Hazards Model with a Diverging Number of Parameters

Regularized variable selection is a powerful tool for identifying the true regression model from a large number of candidates by applying penalties to the objective functions. The penalty functions typically involve a tuning parameter that controls the complexity of the selected model. The ability of the regularized variable selection methods to identify the true model critically depends on the correct choice of the tuning parameter. In this study, we develop a consistent tuning parameter selection method for regularized Cox's proportional hazards model with a diverging number of parameters. The tuning parameter is selected by minimizing the generalized information criterion. We prove that, for any penalty that possesses the oracle property, the proposed tuning parameter selection method identifies the true model with probability approaching one as sample size increases. Its finite sample performance is evaluated by simulations. Its practical use is demonstrated in The Cancer Genome Atlas breast cancer data.     

Lasso penalized semiparametric regression on high-dimensional recurrent event data via coordinate descent

This paper studies a fast computational algorithm for variable selection on high-dimensional recurrent event data. Based on the lasso penalized partial likelihood function for the response process of recurrent event data, a coordinate descent algorithm is used to accelerate the estimation of regression coefficients. This algorithm is capable of selecting important predictors for underdetermined problems where the number of predictors far exceeds the number of cases. The selection strength is controlled by a tuning constant that is determined by a generalized cross-validation method. Our numerical experiments on simulated and real data demonstrate the good performance of penalized regression in model building for recurrent event data in high-dimensional settings.     

Robust logistic regression modelling via the elastic net-type regularization and tuning parameter selection

The penalized logistic regression is a useful tool for classifying samples and feature selection. Although the methodology has been widely used in various fields of research, their performance takes a sudden turn for the worst in the presence of outlier, since the logistic regression is based on the maximum log-likelihood method which is sensitive to outliers. It implies that we cannot accurately classify samples and find important factors having crucial information for classification. To overcome the problem, we propose a robust penalized logistic regression based on a weighted likelihood methodology. We also derive an information criterion for choosing the tuning parameters, which is a vital matter in robust penalized logistic regression modelling in line with generalized information criteria. We demonstrate through Monte Carlo simulations and real-world example that the proposed robust modelling strategies perform well for sparse logistic regression modelling even in the presence of outliers.     

Optimal bandwidth matrices in functional principal component analysis of density functions

In order to explore and compare a finite number T of data sets by applying functional principal component analysis (FPCA) to the T associated probability density functions, we estimate these density functions by using the multivariate kernel method. The data set sizes being fixed, we study the behaviour of this FPCA under the assumption that all the bandwidth matrices used in the estimation of densities are proportional to a common parameter h and proportional to either the variance matrices or the identity matrix. In this context, we propose a selection criterion of the parameter h which depends only on the data and the FPCA method. Then, on simulated examples, we compare the quality of approximation of the FPCA when the bandwidth matrices are selected using either the previous criterion or two other classical bandwidth selection methods, that is, a plug-in or a cross-validation method.     

Estimating the Expectation of the Log-Likelihood with Censored Data for Estimator Selection

A criterion for choosing an estimator in a family of semi-parametric estimators from incomplete data is proposed. This criterion is the expected observed log-likelihood (ELL). Adapted versions of this criterion in case of censored data and in presence of explanatory variables are exhibited. We show that likelihood cross-validation (LCV) is an estimator of ELL and we exhibit three bootstrap estimators. A simulation study considering both families of kernel and penalized likelihood estimators of the hazard function (indexed on a smoothing parameter) demonstrates good results of LCV and a bootstrap estimator called ELL_bboot . We apply the ELL_bboot criterion to compare the kernel and penalized likelihood estimators to estimate the risk of developing dementia for women using data from a large cohort study.     

Nonparametric modeling of the gap time in recurrent event data

Recurrent event data arise in many biomedical and engineering studies when failure events can occur repeatedly over time for each study subject. In this article, we are interested in nonparametric estimation of the hazard function for gap time. A penalized likelihood model is proposed to estimate the hazard as a function of both gap time and covariate. Method for smoothing parameter selection is developed from subject-wise cross-validation. Confidence intervals for the hazard function are derived using the Bayes model of the penalized likelihood. An eigenvalue analysis establishes the asymptotic convergence rates of the relevant estimates. Empirical studies are performed to evaluate various aspects of the method. The proposed technique is demonstrated through an application to the well-known bladder tumor cancer data.     

Choice between Semi-parametric Estimators of Markov and Non-Markov Multi-state Models from Coarsened Observations

Abstract.  We consider models based on multivariate counting processes, including multi-state models. These models are specified semi-parametrically by a set of functions and real parameters. We consider inference for these models based on coarsened observations, focusing on families of smooth estimators such as produced by penalized likelihood. An important issue is the choice of model structure, for instance, the choice between a Markov and some non-Markov models. We define in a general context the expected Kullback–Leibler criterion and we show that the likelihood-based cross-validation (LCV) is a nearly unbiased estimator of it. We give a general form of an approximate of the leave-one-out LCV. The approach is studied by simulations, and it is illustrated by estimating a Markov and two semi-Markov illness–death models with application on dementia using data of a large cohort study.     

On the Optimality of Prediction-based Selection Criteria and the Convergence Rates of Estimators

Several estimators of squared prediction error have been suggested for use in model and bandwidth selection problems. Among these are cross-validation, generalized cross-validation and a number of related techniques based on the residual sum of squares. For many situations with squared error loss, e.g. nonparametric smoothing, these estimators have been shown to be asymptotically optimal in the sense that in large samples the estimator minimizing the selection criterion also minimizes squared error loss. However, cross-validation is known not to be asymptotically optimal for some `easy' location problems. We consider selection criteria based on estimators of squared prediction risk for choosing between location estimators. We show that criteria based on adjusted residual sum of squares are not asymptotically optimal for choosing between asymptotically normal location estimators that converge at rate n ^1/2but are when the rate of convergence is slower. We also show that leave-one-out cross-validation is not asymptotically optimal for choosing between √ n -differentiable statistics but leave- d -out cross-validation is optimal when d ∞ at the appropriate rate.     

Model averaging by jackknife criterion for varying-coefficient partially linear models

Abstract

This paper is concerned with model averaging procedure for varying-coefficient partially linear models. We proposed a jackknife model averaging method that involves minimizing a leave-one-out cross-validation criterion, and developed a computational shortcut to optimize the cross-validation criterion for weight choice. The resulting model average estimator is shown to be asymptotically optimal in terms of achieving the smallest possible squared error. The simulation studies have provided evidence of the superiority of the proposed procedures. Our approach is further applied to a real data.     

Variable selection in joint location and scale models of the skew-normal distribution

A regression model with skew-normal errors provides a useful extension for ordinary normal regression models when the data set under consideration involves asymmetric outcomes. Variable selection is an important issue in all regression analyses, and in this paper, we investigate the simultaneously variable selection in joint location and scale models of the skew-normal distribution. We propose a unified penalized likelihood method which can simultaneously select significant variables in the location and scale models. Furthermore, the proposed variable selection method can simultaneously perform parameter estimation and variable selection in the location and scale models. With appropriate selection of the tuning parameters, we establish the consistency and the oracle property of the regularized estimators. Simulation studies and a real example are used to illustrate the proposed methodologies.