Statistical Inference for Partially Linear Regression Models with Measurement Errors

In this paper,the authors investigate three aspects of statistical inference for the partially linear regression models where some covariates are measured with errors.Firstly, a bandwidth selection procedure is proposed,which is a combination of the differencebased technique and GCV method.Secondly,a goodness-of-fit test procedure is proposed, which is an extension of the generalized likelihood technique.Thirdly,a variable selection procedure for the parametric part is provided based on the nonconcave penalization and corrected profile least squares.Same as＂Variable selection via nonconcave penalized likelihood and its oracle properties＂（J.Amer.Statist.Assoc.,96,2001,1348-1360）,it is shown that the resulting estimator has an oracle property with a proper choice of regularization parameters and penalty function.Simulation studies are conducted to illustrate the finite sample performances of the proposed procedures.     

Variable selection for single-index varying-coefficient model

We consider the problem of variable selection for single-index varying-coefficient model, and present a regularized variable selection procedure by combining basis function approximations with SCAD penalty. The proposed procedure simultaneously selects significant covariates with functional coefficients and local significant variables with parametric coefficients. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the estimators are established. The proposed method can naturally be applied to deal with pure single-index model and varying-coefficient model. Finite sample performances of the proposed method are illustrated by a simulation study and the real data analysis.     

Marginal quantile regression for varying coefficient models with longitudinal data

In this paper, we investigate the quantile varying coefficient model for longitudinal data, where the unknown nonparametric functions are approximated by polynomial splines and the estimators are obtained by minimizing the quadratic inference function. The theoretical properties of the resulting estimators are established, and they achieve the optimal convergence rate for the nonparametric functions. Since the objective function is non-smooth, an estimation procedure is proposed that uses induced smoothing and we prove that the smoothed estimator is asymptotically equivalent to the original estimator. Moreover, we propose a variable selection procedure based on the regularization method, which can simultaneously estimate and select important nonparametric components and has the asymptotic oracle property. Extensive simulations and a real data analysis show the usefulness of the proposed method.

     

Robust variable selection for finite mixture regression models

Finite mixture regression (FMR) models are frequently used in statistical modeling, often with many covariates with low significance. Variable selection techniques can be employed to identify the covariates with little influence on the response. The problem of variable selection in FMR models is studied here. Penalized likelihood-based approaches are sensitive to data contamination, and their efficiency may be significantly reduced when the model is slightly misspecified. We propose a new robust variable selection procedure for FMR models. The proposed method is based on minimum-distance techniques, which seem to have some automatic robustness to model misspecification. We show that the proposed estimator has the variable selection consistency and oracle property. The finite-sample breakdown point of the estimator is established to demonstrate its robustness. We examine small-sample and robustness properties of the estimator using a Monte Carlo study. We also analyze a real data set.     

Efficient estimation and variable selection for infinite variance autoregressive models

In this paper, a self-weighted composite quantile regression estimation procedure is developed to estimate unknown parameter in an infinite variance autoregressive (IVAR) model. The proposed estimator is asymptotically normal and more efficient than a single quantile regression estimator. At the same time, the adaptive least absolute shrinkage and selection operator (LASSO) for variable selection are also suggested. We show that the adaptive LASSO based on the self-weighted composite quantile regression enjoys the oracle properties. Simulation studies and a real data example are conducted to examine the performance of the proposed approaches.     

Variable Selection for Partially Linear Models via Adaptive LASSO

Partially linear model is a class of commonly used semiparametric models, this paper focus on variable selection and parameter estimation for partially linear models via adaptive LASSO method. Firstly, based on profile least squares and adaptive LASSO method, the adaptive LASSO estimator for partially linear models are constructed, and the selections of penalty parameter and bandwidth are discussed. Under some regular conditions, the consistency and asymptotic normality for the estimator are investigated, and it is proved that the adaptive LASSO estimator has the oracle properties. The proposed method can be easily implemented. Finally a Monte Carlo simulation study is conducted to assess the finite sample performance of the proposed variable selection procedure, results show the adaptive LASSO estimator behaves well.     

SICA for Cox’s proportional hazards model with a diverging number of parameters

The smooth integration of counting and absolute deviation （SICA） penalized variable selection procedure for high-dimensional linear regression models is proposed by Lv and Fan （2009）. In this article, we extend their idea to Cox＇s proportional hazards （PH） model by using a penalized log partial likelihood with the SICA penalty. The number of the regression coefficients is allowed to grow with the sample size. Based on an approximation to the inverse of the Hessian matrix, the proposed method can be easily carried out with the smoothing quasi-Newton （SQN） algorithm. Under appropriate sparsity conditions, we show that the resulting estimator of the regression coefficients possesses the oracle property. We perform an extensive simulation study to compare our approach with other methods and illustrate it on a well known PBC data for predicting survival from risk factors.     

测量误差模型的自适应LASSO变量选择方法研究

本文研究测量误差模型的自适应LASSO(least absolute shrinkage and selection operator)变量选择和系数估计问题.首先分别给出协变量有测量误差时的线性模型和部分线性模型自适应LASSO参数估计量,在一些正则条件下研究估计量的渐近性质,并且证明选择合适的调整参数,自适应LASSO参数估计量具有oracle性质.其次讨论估计的实现算法及惩罚参数和光滑参数的选择问题.最后通过模拟和一个实际数据分析研究了自适应LASSO变量选择方法的表现,结果表明,变量选择和参数估计效果良好.     

Difference based estimation for partially linear regression models with measurement errors

This paper is concerned with the estimating problem of the partially linear regression models where the linear covariates are measured with additive errors. A difference based estimation is proposed to estimate the parametric component. We show that the resulting estimator is asymptotically unbiased and achieves the semiparametric efficiency bound if the order of the difference tends to infinity. The asymptotic normality of the resulting estimator is established as well. Compared with the corrected profile least squares estimation, the proposed procedure avoids the bandwidth selection. In addition, the difference based estimation of the error variance is also considered. For the nonparametric component, the local polynomial technique is implemented. The finite sample properties of the developed methodology is investigated through simulation studies. An example of application is also illustrated.     

半参数变系数部分线性模型的统计推断

本文在多种复杂数据下, 研究一类半参数变系数部分线性模型的统计推断理论和方法. 首先在纵向数据和测量误差数据等复杂数据下, 研究半参数变系数部分线性模型的经验似然推断问题, 分别提出分组的和纠偏的经验似然方法. 该方法可以有效地处理纵向数据的组内相关性给构造经验似然比函数所带来的困难. 其次在测量误差数据和缺失数据等复杂数据下, 研究模型的变量选择问题, 分别提出一个“纠偏” 的和基于借补值的变量选择方法. 该变量选择方法可以同时选择参数分量及非参数分量中的重要变量, 并且变量选择与回归系数的估计同时进行. 通过选择适当的惩罚参数, 证明该变量选择方法可以相合地识别出真实模型, 并且所得的正则估计具有oracle 性质.     

Sparse estimation in functional linear regression

As a useful tool in functional data analysis, the functional linear regression model has become increasingly common and been studied extensively in recent years. In this paper, we consider a sparse functional linear regression model which is generated by a finite number of basis functions in an expansion of the coefficient function. In this model, we do not specify how many and which basis functions enter the model, thus it is not like a typical parametric model where predictor variables are pre-specified. We study a general framework that gives various procedures which are successful in identifying the basis functions that enter the model, and also estimating the resulting regression coefficients in one-step. We adopt the idea of variable selection in the linear regression setting where one adds a weighted L₁ penalty to the traditional least squares criterion. We show that the procedures in our general framework are consistent in the sense of selecting the model correctly, and that they enjoy the oracle property, meaning that the resulting estimators of the coefficient function have asymptotically the same properties as the oracle estimator which uses knowledge of the underlying model. We investigate and compare several methods within our general framework, via a simulation study. Also, we apply the methods to the Canadian weather data.     

Adaptive Penalized Weighted Least Absolute Deviations Estimation for the Accelerated Failure Time Model

The accelerated failure time model always offers a valuable complement to the traditional Cox proportional hazards model due to its direct and meaningful interpretation. We propose a variable selection method in the context of the accelerated failure time model for survival data, which can simultaneously complete variable selection and parameter estimation. Meanwhile, the proposed method can deal with the potential outliers in survival times as well as heteroscedastic model errors, which are frequently encountered in practice. Specifically, utilizing the general nonconvex penalty, we propose the adaptive penalized weighted least absolute deviation estimator for the accelerated failure time model. Under some regularity conditions, we show that the proposed method yields consistent estimator and possesses the oracle property. In addition, we propose a new algorithm to compute the estimate in the high dimensional settings, and evaluate the practical utility of the proposed method through extensive simulation studies and two real examples.     

Simultaneous estimation and variable selection in median regression using Lasso-type penalty

We consider the median regression with a LASSO-type penalty term for variable selection. With the fixed number of variables in regression model, a two-stage method is proposed for simultaneous estimation and variable selection where the degree of penalty is adaptively chosen. A Bayesian information criterion type approach is proposed and used to obtain a data-driven procedure which is proved to automatically select asymptotically optimal tuning parameters. It is shown that the resultant estimator achieves the so-called oracle property. The combination of the median regression and LASSO penalty is computationally easy to implement via the standard linear programming. A random perturbation scheme can be made use of to get simple estimator of the standard error. Simulation studies are conducted to assess the finite-sample performance of the proposed method. We illustrate the methodology with a real example.     

Some theoretical results on the Grouped Variables Lasso

We consider the linear regression model with Gaussian error. We estimate the unknown parameters by a procedure inspired by the Group Lasso estimator introduced in [22]. We show that this estimator satisfies a sparsity inequality, i.e., a bound in terms of the number of non-zero components of the oracle regression vector. We prove that this bound is better, in some cases, than the one achieved by the Lasso and the Dantzig selector.      

Asymptotic properties of Lasso in high-dimensional partially linear models

We study the properties of the Lasso in the high-dimensional partially linear model where the number of variables in the linear part can be greater than the sample size. We use truncated series expansion based on polynomial splines to approximate the nonparametric component in this model. Under a sparsity assumption on the regression coefficients of the linear component and some regularity conditions, we derive the oracle inequalities for the prediction risk and the estimation error. We also provide sufficient conditions under which the Lasso estimator is selection consistent for the variables in the linear part of the model. In addition, we derive the rate of convergence of the estimator of the nonparametric function. We conduct simulation studies to evaluate the finite sample performance of variable selection and nonparametric function estimation.     

Nonparametric regression estimation with general parametric error covariance

The asymptotic distribution for the local linear estimator in nonparametric regression models is established under a general parametric error covariance with dependent and heterogeneously distributed regressors. A two-step estimation procedure that incorporates the parametric information in the error covariance matrix is proposed. Sufficient conditions for its asymptotic normality are given and its efficiency relative to the local linear estimator is established. We give examples of how our results are useful in some recently studied regression models. A Monte Carlo study confirms the asymptotic theory predictions and compares our estimator with some recently proposed alternative estimation procedures.     

Variable selection for semiparametric varying coefficient partially linear errors-in-variables models

This paper focuses on the variable selections for semiparametric varying coefficient partially linear models when the covariates in the parametric and nonparametric components are all measured with errors. A bias-corrected variable selection procedure is proposed by combining basis function approximations with shrinkage estimations. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the regularized estimators are established. A simulation study and a real data application are undertaken to evaluate the finite sample performance of the proposed method.     

因变量缺失下变系数部分线性测量误差模型的变量选择

主要研究因变量存在缺失且协变量部分包含测量误差情形下,如何对变系数部分线性模型同时进行参数估计和变量选择.我们利用插补方法来处理缺失数据,并结合修正的profile最小二乘估计和SCAD惩罚对参数进行估计和变量选择.并且证明所得的估计具有渐近正态性和Oracle性质.通过数值模拟进一步研究所得估计的有限样本性质.     

Checking the adequacy of partial linear models with missing covariates at random

In this paper, we consider the goodness-of-fit for checking whether the nonparametric function in a partial linear regression model with missing covariate at random is a parametric one or not. We estimate the selection probability by using parametric and nonparametric approaches. Two score type tests are constructed with the estimated selection probability. The asymptotic distributions of the test statistics are investigated under the null and local alterative hypothesis. Simulation studies are carried out to examine the finite sample performance of the sizes and powers of the tests. We apply the proposed procedure to a data set on the AIDS clinical trial group (ACTG 315) study.     

AMlet,RAMlet, and GAMlet: Automatic Nonlinear Fitting of Additive Models,Robust and Generalized,With Wavelets

A simple and yet powerful method is presented to estimate nonlinearly and nonparametrically the components of additive models using wavelets. The estimator enjoys the good statistical and computational properties of the Waveshrink scatterplot smoother and it can be efficiently computed using the block coordinate relaxation optimization technique. A rule for the automatic selection of the smoothing parameters, suitable for data mining of large datasets, is derived. The wavelet-based method is then extended to estimate generalized additive models. A primal-dual log-barrier interior point algorithm is proposed to solve the corresponding convex programming problem. Based on an asymptotic analysis, a rule for selecting the smoothing parameters is derived, enabling the estimator to be fully automated in practice. We illustrate the finite sample property with a Gaussian and a Poisson simulation.