Robust estimation in generalized semiparametric mixed models for longitudinal data

In this paper, we consider robust generalized estimating equations for the analysis of semiparametric generalized partial linear mixed models (GPLMMs) for longitudinal data. We approximate the non-parametric function in the GPLMM by a regression spline, and make use of bounded scores and leverage-based weights in the estimating equation to achieve robustness against outliers and influential data points, respectively. Under some regularity conditions, the asymptotic properties of the robust estimators are investigated. To avoid the computational problems involving high-dimensional integrals in our estimators, we adopt a robust Monte Carlo Newton-Raphson (RMCNR) algorithm for fitting GPLMMs. Small simulations are carried out to study the behavior of the robust estimates in the presence of outliers, and these estimates are also compared to their corresponding non-robust estimates. The proposed robust method is illustrated in the analysis of two real data sets.     

纵向数据的有效秩推断基于修正的Cholesky分解

本文基于修正的Cholesky分解提出新的方法估计纵向秩回归的组内协方差矩阵,进而提出新的无偏估计函数改善不平衡纵向数据的估计效率.在一些正则条件下,建立了所提估计的渐近正态性.进一步,提出稳健的秩得分检验统计量对回归系数做假设检验.模拟研究和实证分析表明所提方法能够获得高度有效的估计以及所提检验方法比存在的方法更好.     

纵向数据下广义估计方程估计

广义估计方程方法是一种最一般的参数估计方法,广泛地应用于生物统计、经济计量、医疗保险等领域.在纵向数据下,由于组间数据是相关的,为了提高估计的效率,广义估计方程方法一般需要考虑个体组内相关性.因此,大多数文献对个体组内的协方差矩阵进行参数假设,但假设的合理性及协方差矩阵估计的好坏对参数估计效率产生很大影响,同时参数假设也可能导致模型误判.针对纵向数据下广义估计方程,本文提出了改进的GMM方法和经验似然方法,并对给出的估计量建立了大样本性质.其中分块的思想,避免了对个体组内相关性结构进行假设,从这种意义上说,这种方法具有一定的稳健性.我们还通过两个模拟的例子,考察了文中提出估计量的有限样本性质.     

Kernel Smoothing Estimation for Varying Coefficient EV Models with Longitudinal Data

Varying coefficient EV models with longitudinal data are considered. The local bias-corrected kernel estimators for the unknown coefficient functions are proposed. It is shown that the proposed estimators are asymptotically normal under some suitable conditions, and hence it can be used to construct the pointwise confidence regions of the coefficient functions. The finite-sample properties of the proposed procedures are studied through a simulation study.     

Smoothing combined generalized estimating equations in quantile partially linear additive models with longitudinal data

This paper develops a robust and efficient estimation procedure for quantile partially linear additive models with longitudinal data, where the nonparametric components are approximated by B spline basis functions. The proposed approach can incorporate the correlation structure between repeated measures to improve estimation efficiency. Moreover, the new method is empirically shown to be much more efficient and robust than the popular generalized estimating equations method for non-normal correlated random errors. However, the proposed estimating functions are non-smooth and non-convex. In order to reduce computational burdens, we apply the induced smoothing method for fast and accurate computation of the parameter estimates and its asymptotic covariance. Under some regularity conditions, we establish the asymptotically normal distribution of the estimators for the parametric components and the convergence rate of the estimators for the nonparametric functions. Furthermore, a variable selection procedure based on smooth-threshold estimating equations is developed to simultaneously identify non-zero parametric and nonparametric components. Finally, simulation studies have been conducted to evaluate the finite sample performance of the proposed method, and a real data example is analyzed to illustrate the application of the proposed method.     

Robust estimation in joint mean–covariance regression model for longitudinal data

In this paper, we develop robust estimation for the mean and covariance jointly for the regression model of longitudinal data within the framework of generalized estimating equations (GEE). The proposed approach integrates the robust method and joint mean–covariance regression modeling. Robust generalized estimating equations using bounded scores and leverage-based weights are employed for the mean and covariance to achieve robustness against outliers. The resulting estimators are shown to be consistent and asymptotically normally distributed. Simulation studies are conducted to investigate the effectiveness of the proposed method. As expected, the robust method outperforms its non-robust version under contaminations. Finally, we illustrate by analyzing a hormone data set. By downweighing the potential outliers, the proposed method not only shifts the estimation in the mean model, but also shrinks the range of the innovation variance, leading to a more reliable estimation in the covariance matrix.     

基于纵向数据的半参数变系数部分线性回归模型

纵向数据是数理统计研究中的复杂数据类型之一0,在生物、医学和经济学中具有广泛的应用．在实际中经常需要对纵向数据进行统计分析和建模．文章讨论了纵向数据下的半参数变系数部分线性回归模型,这里的纵向数据的在纵向观察在时间上可以是不均等的,也可看成是按某一随机过程来发生．所研究的半参数变系数模型包括了许多半参数模型,比如部分线性模型和变系数模型等．利用计数过程理论和局部线性回归方法,对于纵向数据下半参数变系数进行了统计推断,给出了参数分量和非参数分量的profile最小二乘估计,研究了这些估计的渐近性质,获得这些估计的相合性和渐近正态性．     

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In this article, we develop efficient robust method for estimation of mean and covariance simultaneously for longitudinal data in regression model. Based on Cholesky decomposition for the covariance matrix and rewriting the regression model, we propose a weighted least square estimator, in which the weights are estimated under generalized empirical likelihood framework. The proposed estimator obtains high efficiency from the close connection to empirical likelihood method, and achieves robustness by bounding the weighted sum of squared residuals. Simulation study shows that, compared to existing robust estimation methods for longitudinal data, the proposed estimator has relatively high efficiency and comparable robustness. In the end, the proposed method is used to analyse a real data set.     

ASYMPTOTIC PROPERTIES OF ESTIMATORS IN PARTIALLY LINEAR SINGLE-INDEX MODEL FOR LONGITUDINAL DATA

In this article, a partially linear single-index model /or longitudinal data is investigated. The generalized penalized spline least squares estimates of the unknown parameters are suggested. All parameters can be estimated simultaneously by the proposed method while the feature of longitudinal data is considered. The existence, strong consistency and asymptotic normality of the estimators are proved under suitable conditions. A simulation study is conducted to investigate the finite sample performance of the proposed method. Our approach can also be used to study the pure single-index model for longitudinal data.     

半相依部分线性可加回归模型的统计推断

受实际问题研究的启发, 为减少模型偏差, 提出了一类半相依部分线性可加的半参数回归模型. 这类半相依模型中, 响应变量与 一部分解释变量之间的关系是线性的, 与另一部分解释变量之间的关系未知但具有可加结构, 各方程的误差之间是相关的. 将级 数逼近法、最小二乘法和同期相关的估计结合起来, 提出了用于估计模型参数分量的加权半参数最小二乘估计量(WSLSEs), 和用于估 计模型非参数分量的加权级数逼近估计量(WSEs). 证明了这些加权的估计量比相应的不加权的估计量渐近有效, 并导出了相应的渐近正态性. 另外, 还讨论了利用这些估计量的渐近性质来对模型的参数及非参数分量作统计推断. 用大量的模拟实验考察 了所提出的方法在有限样本情况下的表现, 并对美国的一个关于妇女工资问题的全国纵向调查(NLS)数据集进行了统计分析.     

Semiparametric analysis of longitudinal data with informative observation times

In many longitudinal studies,observation times as well as censoring times may be correlated with longitudinal responses.This paper considers a multiplicative random effects model for the longitudinal response where these correlations may exist and a joint modeling approach is proposed via a shared latent variable.For inference about regression parameters,estimating equation approaches are developed and asymptotic properties of the proposed estimators are established.The finite sample behavior of the methods is examined through simulation studies and an application to a data set from a bladder cancer study is provided for illustration.     

部分线性混合效应模型中方差分量的稳健估计

部分线性混合效应模型中方差分量是我们感兴趣的参数, 文献中已经给出许多估计方法. 但是其中很多方法都可以归结为广义估计方程方法(GEE), 如: 最大似然估计(MLE), 约束最大似然估计(REMLE)等, 而GEE方法对异常点很敏感. 本文提出一组关于部分线性混合效应模型(PLMM)中均值和方差分量的稳健估计方程, 对均值和方差分量同时进行稳健估计; 并进行了随机模拟考察所提出稳健估计的有效性, 最后通过两个实例, 说明了所提方法的可行性.     

Mixtures of semiparametric varying coefficient models for longitudinal data with nonignorable dropout

Informative dropout often arise in longitudinal data. In this paper we propose a mixture model in which the responses follow a semiparametric varying coefficient random effects model and some of the regression coefficients depend on the dropout time in a non-parametric way. The local linear version of the profile-kernel method is used to estimate the parameters of the model. The proposed estimators are shown to be consistent and asymptotically normal, and the finite performance of the estimators is evaluated by numerical simulation.     

Multivariate generalized S-estimators

In this paper we introduce generalized S-estimators for the multivariate regression model. This class of estimators combines high robustness and high efficiency. They are defined by minimizing the determinant of a robust estimator of the scatter matrix of differences of residuals. In the special case of a multivariate location model, the generalized S-estimator has the important independence property, and can be used for high breakdown estimation in independent component analysis. Robustness properties of the estimators are investigated by deriving their breakdown point and the influence function. We also study the efficiency of the estimators, both asymptotically and at finite samples. To obtain inference for the regression parameters, we discuss the fast and robust bootstrap for multivariate generalized S-estimators. The method is illustrated on a real data example.     

Variable selection in the partially linear errors-in-variables models for longitudinal data

This paper proposes a new approach for variable selection in partially linear errors-in-variables (EV) models for longitudinal data by penalizing appropriate estimating functions. We apply the SCAD penalty to simultaneously select significant variables and estimate unknown parameters. The rate of convergence and the asymptotic normality of the resulting estimators are established. Furthermore, with proper choice of regularization parameters, we show that the proposed estimators perform as well as the oracle procedure. A new algorithm is proposed for solving penalized estimating equation. The asymptotic results are augmented by a simulation study.     

对稳健回归尺度参数估计的一种改进

常对线性回归模型的稳健 M估计中 ,尺度参数使用绝对离差中位数 MAD.将 Rousseeuw等人对单变量尺度参数的一种稳健估计 Sn引入到回归问题中 ,讨论了此估计的一些优良性质 ,并通过一个小规模的模拟研究 ,说明使用 Sn比使用 MAD做尺度参数将会较大地提高回归估计的估计效率 .     

Efficient parameter estimation via modified Cholesky decomposition for quantile regression with longitudinal data

It is well known that specifying a covariance matrix is difficult in the quantile regression with longitudinal data. This paper develops a two step estimation procedure to improve estimation efficiency based on the modified Cholesky decomposition. Specifically, in the first step, we obtain the initial estimators of regression coefficients by ignoring the possible correlations between repeated measures. Then, we apply the modified Cholesky decomposition to construct the covariance models and obtain the estimator of within-subject covariance matrix. In the second step, we construct unbiased estimating functions to obtain more efficient estimators of regression coefficients. However, the proposed estimating functions are discrete and non-convex. We utilize the induced smoothing method to achieve the fast and accurate estimates of parameters and their asymptotic covariance. Under some regularity conditions, we establish the asymptotically normal distributions for the resulting estimators. Simulation studies and the longitudinal progesterone data analysis show that the proposed approach yields highly efficient estimators.     

Lower Rank Approximation of Matrices Based on Fast and Robust Alternating Regression

We introduce fast and robust algorithms for lower rank approximation to given matrices based on robust alternating regression. The alternating least squares regression, also called criss-cross regression, was used for lower rank approximation of matrices, but it lacks robustness against outliers in these matrices. We use robust regression estimators and address some of the complications arising from this approach. We find it helpful to use high breakdown estimators in the initial iterations, followed by M estimators with monotone score functions in later iterations towards convergence. In addition to robustness, the computational speed is another important consideration in the development of our proposed algorithm, because alternating robust regression can be computationally intensive for large matrices. Based on a mix of the least trimmed squares (LTS) and Huber's M estimators, we demonstrate that fast and robust lower rank approximations are possible for modestly large matrices.     

Asymptotic distributions of nonparametric regression estimators for longitudinal or functional data

The estimation of a regression function by kernel method for longitudinal or functional data is considered. In the context of longitudinal data analysis, a random function typically represents a subject that is often observed at a small number of time points, while in the studies of functional data the random realization is usually measured on a dense grid. However, essentially the same methods can be applied to both sampling plans, as well as in a number of settings lying between them. In this paper general results are derived for the asymptotic distributions of real-valued functions with arguments which are functionals formed by weighted averages of longitudinal or functional data. Asymptotic distributions for the estimators of the mean and covariance functions obtained from noisy observations with the presence of within-subject correlation are studied. These asymptotic normality results are comparable to those standard rates obtained from independent data, which is illustrated in a simulation study. Besides, this paper discusses the conditions associated with sampling plans, which are required for the validity of local properties of kernel-based estimators for longitudinal or functional data.     

线性模型参数的稳健化有偏估计

本文讨论复共线性和粗差同时存在时线性模型的参数估计问题,基于等价权原理提出了一个稳健有偏估计类（稳健压缩估计）,并且建立了稳健压缩估计的计算方法,为了满足实际问题的需要,构造了许多很有意义的稳健有偏估计,例如稳健岭估计、稳健主成分估计,稳健组合主成估计、稳健单参数主成分估计、稳健根方估计等等,最后通过一个算例表明,本文提出的稳健有偏估计具有既可克服复共线性影响又可抵抗粗差干扰的良好性质。