Sparsity identification for high-dimensional partially linear model with measurement error

In this article, we studied the identification of significant predictors in partially linear model in which some regressors are contaminated with random errors. Moreover, the dimension of parametric component is divergent and the regression coefficients are sparse. We applied difference technique to remove the nonparametric component for circumventing the selection of bandwidth, and constructed a bias-corrected shrinking estimator for the coefficient by using smoothly clipped absolute deviation (SCAD) penalty. Then, we derived the estimating and selecting consistency and established the asymptotic distribution for the identified significant estimators. Finally, Monte Carlo studies illustrate the performance of our approach.     

On preliminary test ridge regression estimators for linear restrictions in a regression model with non-normal disturbances

In this paper, we study the properties of the preliminary test, restricted and unrestricted ridge regression estimators of the linear regression model with non-normal disturbances. We present the estimators of the regression coefficients combining the idea of preliminary test and ridge regression methodology, when it is suspected that the regression coefficients may be restricted to a subspace and the regression error is distributed as multivariate t. Accordingly we consider three estimators, namely the Unrestricted Ridge Regression Estimator (URRRE), the Restricted Ridge Regression Estimator (RRRE) and finally the Preliminary test Ridge Regression Estimator (PTRRE). The biases and the mean square error (MSE) of the estimators are derived under the null and alternative hypotheses and compared with the usual estimators. By studying the MSE criterion, the regions of optimahty of the estimators are determined.     

Nonparametric Shrinkage Estimation for Aalen's Additive Hazards Model

Aalen's nonparametric additive model in which the regression coefficients are assumed to be unspecified functions of time is a flexible alternative to Cox's proportional hazards model when the proportionality assumption is in doubt. In this paper, we incorporate a general linear hypothesis into the estimation of the time‐varying regression coefficients. We combine unrestricted least squares estimators and estimators that are restricted by the linear hypothesis and produce James‐Stein‐type shrinkage estimators of the regression coefficients. We develop the asymptotic joint distribution of such restricted and unrestricted estimators and use this to study the relative performance of the proposed estimators via their integrated asymptotic distributional risks. We conduct Monte Carlo simulations to examine the relative performance of the estimators in terms of their integrated mean square errors. We also compare the performance of the proposed estimators with a recently devised LASSO estimator as well as with ridge‐type estimators both via simulations and data on the survival of primary billiary cirhosis patients.     

Performance of some new preliminary test ridge regression estimators and their properties

The problem of estimation of the regression coefficients in a multiple regression model is considered under multicollinearity situation when it is suspected that the regression coefficients may be restricted to a subspace. We present the estimators of the regression coefficients combining the idea of preliminary test and ridge regression methodology. Accordingly, we consider three estimators, namely, the unrestricted ridge regression estimator (URRE), the restricted ridge regression estimator (RRRE), and finally, the preliminary test ridge regression estimator (PTRRE). The biases, variancematrices and mean square errors (mse) of the estimators are derived and compared with the usual estimators. Regions of optimality of the estimators are determined by studying the mse criterion. The conditions of superiority of the estimators over the traditional estimators as in Saleh and Han (1990) and Ali and Saleh (1991) have also been discussed.     

Semiparametric statistical inferences for longitudinal data with nonparametric covariance modelling

In this paper, semiparametric modelling for longitudinal data with an unstructured error process is considered. We propose a partially linear additive regression model for longitudinal data in which within-subject variances and covariances of the error process are described by unknown univariate and bivariate functions, respectively. We provide an estimating approach in which polynomial splines are used to approximate the additive nonparametric components and the within-subject variance and covariance functions are estimated nonparametrically. Both the asymptotic normality of the resulting parametric component estimators and optimal convergence rate of the resulting nonparametric component estimators are established. In addition, we develop a variable selection procedure to identify significant parametric and nonparametric components simultaneously. We show that the proposed SCAD penalty-based estimators of non-zero components have an oracle property. Some simulation studies are conducted to examine the finite-sample performance of the proposed estimation and variable selection procedures. A real data set is also analysed to demonstrate the usefulness of the proposed method.     

Estimation and variable selection in partial linear single index models with error-prone linear covariates

We study the estimation and variable selection for a partial linear single index model (PLSIM) when some linear covariates are not observed, but their ancillary variables are available. We use the semiparametric profile least-square based estimation procedure to estimate the parameters in the PLSIM after the calibrated error-prone covariates are obtained. Asymptotic normality for the estimators are established. We also employ the smoothly clipped absolute deviation (SCAD) penalty to select the relevant variables in the PLSIM. The resulting SCAD estimators are shown to be asymptotically normal and have the oracle property. Performance of our estimation procedure is illustrated through numerous simulations. The approach is further applied to a real data example.     

Estimation and Variable Selection for Semiparametric Additive Partial Linear Models (SS-09-140)

Semiparametric additive partial linear models, containing both linear and nonlinear additive components, are more flexible compared to linear models, and they are more efficient compared to general nonparametric regression models because they reduce the problem known as "curse of dimensionality". In this paper, we propose a new estimation approach for these models, in which we use polynomial splines to approximate the additive nonparametric components and we derive the asymptotic normality for the resulting estimators of the parameters. We also develop a variable selection procedure to identify significant linear components using the smoothly clipped absolute deviation penalty (SCAD), and we show that the SCAD-based estimators of non-zero linear components have an oracle property. Simulations are performed to examine the performance of our approach as compared to several other variable selection methods such as the Bayesian Information Criterion and Least Absolute Shrinkage and Selection Operator (LASSO). The proposed approach is also applied to real data from a nutritional epidemiology study, in which we explore the relationship between plasma beta-carotene levels and personal characteristics (e.g., age, gender, body mass index (BMI), etc.) as well as dietary factors (e.g., alcohol consumption, smoking status, intake of cholesterol, etc.).     

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.     

Robust variable selection and parametric component identification in varying coefficient models

ABSTRACT

In this paper, we study a novelly robust variable selection and parametric component identification simultaneously in varying coefficient models. The proposed estimator is based on spline approximation and two smoothly clipped absolute deviation (SCAD) penalties through rank regression, which is robust with respect to heavy-tailed errors or outliers in the response. Furthermore, when the tuning parameter is chosen by modified BIC criterion, we show that the proposed procedure is consistent both in variable selection and the separation of varying and constant coefficients. In addition, the estimators of varying coefficients possess the optimal convergence rate under some assumptions, and the estimators of constant coefficients have the same asymptotic distribution as their counterparts obtained when the true model is known. Simulation studies and a real data example are undertaken to assess the finite sample performance of the proposed variable selection procedure.     

Robust estimation and variable selection for varying-coefficient single-index models based on modal regression

ABSTRACT

In this paper, we propose a new efficient and robust penalized estimating procedure for varying-coefficient single-index models based on modal regression and basis function approximations. The proposed procedure simultaneously solves two types of problems: separation of varying and constant effects and selection of variables with non zero coefficients for both non parametric and index components using three smoothly clipped absolute deviation (SCAD) penalties. With appropriate selection of the tuning parameters, the new method possesses the consistency in variable selection and the separation of varying and constant coefficients. In addition, the estimators of varying coefficients possess the optimal convergence rate and the estimators of constant coefficients and index parameters have the oracle property. Finally, we investigate the finite sample performance of the proposed method through a simulation study and real data analysis.     

Risk Comparison of Inequality Constrained Estimators in the Heteroscedastic Linear Model

There exist many studies which treat the inequality and/or interval constraints on coefficients in the homoscedastic linear regression model. However, the sampling performance of the inequality constrained estimators in the heteroscedastic linear model has not been examined. This paper considers the inequality constrained estimators in the heteroscedastic linear regression model and derives their risks under a quadratic loss function. Furthermore, using the inequality constrained estimators, we introduce a pre-test estimator which might be employed after the test for homoscedasticity and derive its risk. In addition, the risk performance of these estimators is evaluated numerically.     

Shrinkage and Penalty Estimators of a Poisson Regression Model

In this paper we propose Stein‐type shrinkage estimators for the parameter vector of a Poisson regression model when it is suspected that some of the parameters may be restricted to a subspace. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Furthermore, we consider three different penalty estimators: the LASSO, adaptive LASSO, and SCAD estimators and compare their relative performance with that of the shrinkage estimators. Monte Carlo simulation studies reveal that the shrinkage strategy compares favorably to the use of penalty estimators, in terms of relative mean squared error, when the number of inactive predictors in the model is moderate to large. The shrinkage and penalty strategies are applied to two real data sets to illustrate the usefulness of the procedures in practice.     

The exact density and distribution functions of the inequality constrained and pre-test estimators

We consider a bivariate normal linear regression model with an inequality restriction imposed on one of the regression coefficients. The exact analytical expressions for the density and distribution functions of the inequality constrained and pre-test estimators are derived and numerically evaluated. The implications of using the inequality constrained and pre-test estimators in confidence interval construction are also discussed and explored.     

Partial linear single-index models with additive distortion measurement errors

We study partial linear single-index models (PLSiMs) when the response and the covariates in the parametric part are measured with additive distortion measurement errors. These distortions are modeled by unknown functions of a commonly observable confounding variable. We use the semiparametric profile least-squares method to estimate the parameters in the PLSiMs based on the residuals obtained from the distorted variables and confounding variable. We also employ the smoothly clipped absolute deviation penalty (SCAD) to select the relevant variables in the PLSiMs. We show that the resulting SCAD estimators are consistent and possess the oracle property. For the non parametric link function, we construct the simultaneous confidence bands and obtain the asymptotic distribution of the maximum absolute deviation between the estimated link function and the true link function. A simulation study is conducted to evaluate the performance of the proposed methods and a real dataset is analyzed for illustration.     

Model selection and model averaging for semiparametric partially linear models with missing data

We study model selection and model averaging in semiparametric partially linear models with missing responses. An imputation method is used to estimate the linear regression coefficients and the nonparametric function. We show that the corresponding estimators of the linear regression coefficients are asymptotically normal. Then a focused information criterion and frequentist model average estimators are proposed and their theoretical properties are established. Simulation studies are performed to demonstrate the superiority of the proposed methods over the existing strategies in terms of mean squared error and coverage probability. Finally, the approach is applied to a real data case.     

Use of prior information in the form of interval constraints for the improved estimation of linear regression models with some missing responses

We consider the estimation of coefficients in a linear regression model when some responses on the study variable are missing and some prior information in the form of lower and upper bounds for the average values of missing responses is available. Employing the mixed regression framework, we present five estimators for the vector of regression coefficients. Their exact as well as asymptotic properties are discussed and superiority of one estimator over the other is examined.     

Quadratic Approximation via the SCAD Penalty with a Diverging Number of Parameters

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 the smoothly 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.     

Replicated measurement error model under exact linear restrictions

We consider a replicated ultrastructural measurement error regression model where predictor variables are observed with error. It is assumed that some prior information regarding the regression coefficients is available in the form of exact linear restrictions. Three classes of estimators of regression coefficients are proposed. These estimators are shown to be consistent as well as satisfying the given restrictions. The asymptotic properties of unrestricted as well as restricted estimators are studied without imposing any distributional assumption on any random component of the model. A Monte Carlo simulations study is performed to assess the effect of sample size, replicates and non-normality on the estimators.     

A class of Stein‐rules in multivariate regression model with structural changes

In this paper, we consider an estimation problem of the matrix of the regression coefficients in multivariate regression models with unknown change‐points. More precisely, we consider the case where the target parameter satisfies an uncertain linear restriction. Under general conditions, we propose a class of estimators that includes as special cases shrinkage estimators (SEs) and both the unrestricted and restricted estimator. We also derive a more general condition for the SEs to dominate the unrestricted estimator. To this end, we extend some results underlying the multidimensional version of the mixingale central limit theorem as well as some important identities for deriving the risk function of SEs. Finally, we present some simulation studies that corroborate the theoretical findings.     

Model averaging procedure for varying-coefficient partially linear models with missing responses

This paper is concerned with model averaging procedure for varying-coefficient partially linear models with missing responses. The profile least-squares estimation process and inverse probability weighted method are employed to estimate regression coefficients of the partially restricted models, in which the propensity score is estimated by the covariate balancing propensity score method. The estimators of the linear parameters are shown to be asymptotically normal. Then we develop the focused information criterion, formulate the frequentist model averaging estimators and construct the corresponding confidence intervals. Some simulation studies are conducted to examine the finite sample performance of the proposed methods. We find that the covariate balancing propensity score improves the performance of the inverse probability weighted estimator. We also demonstrate the superiority of the proposed model averaging estimators over those of existing strategies in terms of mean squared error and coverage probability. Finally, our approach is further applied to a real data example.