Difference-based estimation and model identification for panel data semiparametric models with cross-section dependence

Abstract

In this article, we consider a panel data partially linear regression model with fixed effect and non parametric time trend function. The data can be dependent cross individuals through linear regressor and error components. Unlike the methods using non parametric smoothing technique, a difference-based method is proposed to estimate linear regression coefficients of the model to avoid bandwidth selection. Here the difference technique is employed to eliminate the non parametric function effect, not the fixed effects, on linear regressor coefficient estimation totally. Therefore, a more efficient estimator for parametric part is anticipated, which is shown to be true by the simulation results. For the non parametric component, the polynomial spline technique is implemented. The asymptotic properties of estimators for parametric and non parametric parts are presented. We also show how to select informative ones from a number of covariates in the linear part by using smoothly clipped absolute deviation-penalized estimators on a difference-based least-squares objective function, and the resulting estimators perform asymptotically as well as the oracle procedure in terms of selecting the correct model.     

Ridge estimation in generalized linear models and proportional hazards regressions

Conventionally, a ridge parameter is estimated as a function of regression parameters based on ordinary least squares. In this article, we proposed an iterative procedure instead of the one-step or conventional ridge method. Additionally, we construct an indicator that measures the potential degree of improvement in mean squared error when ridge estimates are employed. Simulations show that our methods are appropriate for a wide class of non linear models including generalized linear models and proportional hazards (PHs) regressions. The method is applied to a PH regression with highly collinear covariates in a cancer recurrence study.     

Modelling Survival Events with Longitudinal Covariates Measured with Error

In survival analysis, time-dependent covariates are usually present as longitudinal data collected periodically and measured with error. The longitudinal data can be assumed to follow a linear mixed effect model and Cox regression models may be used for modelling of survival events. The hazard rate of survival times depends on the underlying time-dependent covariate measured with error, which may be described by random effects. Most existing methods proposed for such models assume a parametric distribution assumption on the random effects and specify a normally distributed error term for the linear mixed effect model. These assumptions may not be always valid in practice. In this article, we propose a new likelihood method for Cox regression models with error-contaminated time-dependent covariates. The proposed method does not require any parametric distribution assumption on random effects and random errors. Asymptotic properties for parameter estimators are provided. Simulation results show that under certain situations the proposed methods are more efficient than the existing methods.     

Generalized partially linear varying-coefficient models

Generalized varying-coefficient models are useful extensions of generalized linear models. They arise naturally when investigating how regression coefficients change over different groups characterized by certain covariates such as age. In this paper, we extend these models to generalized partially linear varying-coefficient models, in which some coefficients are constants and the others are functions of certain covariates. Procedures for estimating the linear and non-parametric parts are developed and their associated statistical properties are studied. The methods proposed are illustrated using some simulations and real data analysis.     

Weighted composite quantile regression for partially linear varying coefficient models

Partially linear varying coefficient models (PLVCMs) with heteroscedasticity are considered in this article. Based on composite quantile regression, we develop a weighted composite quantile regression (WCQR) to estimate the non parametric varying coefficient functions and the parametric regression coefficients. The WCQR is augmented using a data-driven weighting scheme. Moreover, the asymptotic normality of proposed estimators for both the parametric and non parametric parts are studied explicitly. In addition, by comparing the asymptotic relative efficiency theoretically and numerically, WCQR method all outperforms the CQR method and some other estimate methods. To achieve sparsity with high-dimensional covariates, we develop a variable selection procedure to select significant parametric components for the PLVCM and prove the method possessing the oracle property. Both simulations and data analysis are conducted to illustrate the finite-sample performance of the proposed methods.     

A class of partially linear single‐index survival models

The authors define a class of “partially linear single‐index” survival models that are more flexible than the classical proportional hazards regression models in their treatment of covariates. The latter enter the proposed model either via a parametric linear form or a nonparametric single‐index form. It is then possible to model both linear and functional effects of covariates on the logarithm of the hazard function and if necessary, to reduce the dimensionality of multiple covariates via the single‐index component. The partially linear hazards model and the single‐index hazards model are special cases of the proposed model. The authors develop a likelihood‐based inference to estimate the model components via an iterative algorithm. They establish an asymptotic distribution theory for the proposed estimators, examine their finite‐sample behaviour through simulation, and use a set of real data to illustrate their approach.     

Cox Regression with Incomplete Covariate Measurements using the EM-algorithm

Ibrahim (1990) used the EM-algorithm to obtain maximum likelihood estimates of the regression parameters in generalized linear models with partially missing covariates. The technique was termed EM by the method of weights. In this paper, we generalize this technique to Cox regression analysis with missing values in the covariates. We specify a full model letting the unobserved covariate values be random and then maximize the observed likelihood. The asymptotic covariance matrix is estimated by the inverse information matrix. The missing data are allowed to be missing at random but also the non-ignorable non-response situation may in principle be considered. Simulation studies indicate that the proposed method is more efficient than the method suggested by Paik & Tsai (1997). We apply the procedure to a clinical trials example with six covariates with three of them having missing values.     

Truncated location-scale non linear regression models

We present a class of truncated non linear regression models for location and scale where the truncated nature of the data is incorporated into the statistical model by assuming that the response variable follows a truncated distribution. The location parameter of the response variable is assumed to be modeled by a continuous non linear function of covariates and unknown parameters. In addition, the proposed model also allows for the scale parameter of the responses to be characterized by a continuous function of the covariates and unknown parameters. Three particular cases of the proposed models are presented by considering the response variable to follow a truncated normal, truncated skew normal, and truncated beta distribution. These truncated non linear regression models are constructed assuming fixed known truncation limits and model parameters are estimated by direct maximization of the log-likelihood using a non linear optimization algorithm. Standardized residuals and diagnostic metrics based on the cases deletion are considered to verify the adequacy of the model and to detect outliers and influential observations. Results based on simulated data are presented to assess the frequentist properties of estimates, and a real data set on soil-water retention from the Buriti Vermelho River Basin database is analyzed using the proposed methodology.     

Modal non‐linear regression in the presence of Laplace measurement error

In this paper, we propose a robust estimation procedure for a class of non‐linear regression models when the covariates are contaminated with Laplace measurement error, aiming at constructing an estimation procedure for the regression parameters which are less affected by the possible outliers, and heavy‐tailed underlying distribution, as well as reducing the bias introduced by the measurement error. Starting with the modal regression procedure developed for the measurement error‐free case, a non‐trivial modification is made so that the modified version can effectively correct the potential bias caused by measurement error. Large sample properties of the proposed estimate, such as the convergence rate and the asymptotic normality, are thoroughly investigated. A simulation study and real data application are conducted to illustrate the satisfying finite sample performance of the proposed estimation procedure.     

Statistical Inference on Partially Linear Additive Models with Missing Response Variables and Error-prone Covariates

This paper considers statistical inference for the partially linear additive models, which are useful extensions of additive models and partially linear models. We focus on the case where some covariates are measured with additive errors, and the response variable is sometimes missing. We propose a profile least-squares estimator for the parametric component and show that the resulting estimator is asymptotically normal. To construct a confidence region for the parametric component, we also propose an empirical-likelihood-based statistic, which is shown to have a chi-squared distribution asymptotically. Furthermore, a simulation study is conducted to illustrate the performance of the proposed methods.     

局部平稳部分线性面板数据模型的统计推断

部分线性模型是一类非常重要的半参数回归模型,由于它既含有参数部分又含有非参数部分,与常规的线性模型相比具有更强的适应性和解释能力。文章研究带有局部平稳协变量的固定效应部分线性面板数据模型的统计推断。首先提出一个两阶段估计方法得到模型中未知参数和非参数函数的估计,并证明估计量的渐近性质,然后运用不变原理构造出非参数函数的一致置信带,最后通过数值模拟研究和实例分析验证了该方法的有效性。     

Penalized inverse probability weighted estimators for weighted rank regression with missing covariates

Abstract

In this article, we study the variable selection and estimation for linear regression models with missing covariates. The proposed estimation method is almost as efficient as the popular least-squares-based estimation method for normal random errors and empirically shown to be much more efficient and robust with respect to heavy tailed errors or outliers in the responses and covariates. To achieve sparsity, a variable selection procedure based on SCAD is proposed to conduct estimation and variable selection simultaneously. The procedure is shown to possess the oracle property. To deal with the covariates missing, we consider the inverse probability weighted estimators for the linear model when the selection probability is known or unknown. It is shown that the estimator by using estimated selection probability has a smaller asymptotic variance than that with true selection probability, thus is more efficient. Therefore, the important Horvitz-Thompson property is verified for penalized rank estimator with the covariates missing in the linear model. Some numerical examples are provided to demonstrate the performance of the estimators.     

Semiparametric mixture of additive regression models

In this article, we propose a semiparametric mixture of additive regression models, in which the regression functions are additive and non parametric while the mixing proportions and variances are constant. Compared with the mixture of linear regression models, the proposed methodology is more flexible in modeling the non linear relationship between the response and covariate. A two-step procedure based on the spline-backfitted kernel method is derived for computation. Moreover, we establish the asymptotic normality of the resultant estimators and examine their good performance through a numerical example.     

Estimation for Partially Linear Single-index Instrumental Variables Models

In this article, we generalize the partially linear single-index models to the scenario with some endogenous covariates variables. It is well known that the estimators based on the existing methods are often inconsistent because of the endogeneity of covariates. To deal with the endogenous variables, we introduce some auxiliary instrumental variables. A three-stage estimation procedure is proposed for partially linear single-index instrumental variables models. The first stage is to obtain a linear projection of endogenous variables on a set of instrumental variables, the second stage is to estimate the link function by using local linear smoother for given constant parameters, and the last stage is to obtain the estimators of constant parameters based on the estimating equation. Asymptotic normality is established for the proposed estimators. Some simulation studies are undertaken to assess the finite sample performance of the proposed estimation procedure.     

NEW EFFICIENT ESTIMATION AND VARIABLE SELECTION METHODS FOR SEMIPARAMETRIC VARYING-COEFFICIENT PARTIALLY LINEAR MODELS

The complexity of semiparametric models poses new challenges to statistical inference and model selection that frequently arise from real applications. In this work, we propose new estimation and variable selection procedures for the semiparametric varying-coefficient partially linear model. We first study quantile regression estimates for the nonparametric varying-coefficient functions and the parametric regression coefficients. To achieve nice efficiency properties, we further develop a semiparametric composite quantile regression procedure. We establish the asymptotic normality of proposed estimators for both the parametric and nonparametric parts and show that the estimators achieve the best convergence rate. Moreover, we show that the proposed method is much more efficient than the least-squares-based method for many non-normal errors and that it only loses a small amount of efficiency for normal errors. In addition, it is shown that the loss in efficiency is at most 11.1% for estimating varying coefficient functions and is no greater than 13.6% for estimating parametric components. To achieve sparsity with high-dimensional covariates, we propose adaptive penalization methods for variable selection in the semiparametric varying-coefficient partially linear model and prove that the methods possess the oracle property. Extensive Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedures. Finally, we apply the new methods to analyze the plasma beta-carotene level data.     

Statistical estimation for partially linear error-in-variable models with error-prone covariates

Motivated by a heart disease data, we propose a new partially linear error-in-variable models with error-prone covariates, in which mismeasured covariate appears in the noparametric part and the covariates in the parametric part are not observed, but ancillary variables are available. In this case, we first calibrate the linear covariates, and then use the least-square method and the local linear method to estimate parametric and nonparametric components. Also, under certain conditions the asymptotic distributions of proposed estimates are obtained. Simulated and real examples are conducted to illustrate our proposed methodology.     

Model selection of generalized partially linear models with missing covariates

In this paper, a generalized partially linear model (GPLM) with missing covariates is studied and a Monte Carlo EM (MCEM) algorithm with penalized-spline (P-spline) technique is developed to estimate the regression coefficients and nonparametric function, respectively. As classical model selection procedures such as Akaike's information criterion become invalid for our considered models with incomplete data, some new model selection criterions for GPLMs with missing covariates are proposed under two different missingness mechanism, say, missing at random (MAR) and missing not at random (MNAR). The most attractive point of our method is that it is rather general and can be extended to various situations with missing observations based on EM algorithm, especially when no missing data involved, our new model selection criterions are reduced to classical AIC. Therefore, we can not only compare models with missing observations under MAR/MNAR settings, but also can compare missing data models with complete-data models simultaneously. Theoretical properties of the proposed estimator, including consistency of the model selection criterions are investigated. A simulation study and a real example are used to illustrate the proposed methodology.     

The em algorithm for the quasi-likelihood regression model

The objective of this paper is to present a method which can accommodate certain types of missing data by using the quasi-likelihood function for the complete data. This method can be useful when we can make first and second moment assumptions only; in addition, it can be helpful when the EM algorithm applied to the actual likelihood becomes overly complicated. First we derive a loss function for the observed data using an exponential family density which has the same mean and variance structure of the complete data. This loss function is the counterpart of the quasi-deviance for the observed data. Then the loss function is minimized using the EM algorithm. The use of the EM algorithm guarantees a decrease in the loss function at every iteration. When the observed data can be expressed as a deterministic linear transformation of the complete data, or when data are missing completely at random, the proposed method yields consistent estimators. Examples are given for overdispersed polytomous data, linear random effects models, and linear regression with missing covariates. Simulation results for the linear regression model with missing covariates show that the proposed estimates are more efficient than estimates based on completely observed units, even when outcomes are bimodal or skewed.     

Testing generalized linear and semiparametric models against smooth alternatives

We propose goodness-of-fit tests for testing generalized linear models and semiparametric regression models against smooth alternatives. The focus is on models having both continous and factorial covariates. As a smooth extension of a parametric or semiparametric model we use generalized varying-coefficient models as proposed by Hastie and Tibshirani. A likelihood ratio statistic is used for testing. Asymptotic expansions allow us to write the estimates as linear smoothers which in turn guarantees simple and fast bootstrapping of the test statistic. The test is shown to have √ n -power, but in contrast with parametric tests it is powerful against smooth alternatives in general.     

A class of partially linear transformation models for recurrent gap times

In this article, we propose a general class of partially linear transformation models for recurrent gap time data, which extends the linear transformation models by incorporating non linear covariate effects and includes the partially linear proportional hazards and the partially linear proportional odds models as special cases. Both global and local estimating equations are developed to estimate the parametric and non parametric covariate effects, and the asymptotic properties of the resulting estimators are established. The finite-sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a clinic study on chronic granulomatous disease is provided.