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.     

纵向数据边际模型非参数光滑方法的比较

对于纵向数据边际模型的均值函数, 有很多非参数估计方法, 其中回归样条, 光滑样条, 似乎不相关(SUR)核估计等方法在工作协方差阵正确指定时具有最小的渐近方差. 回归样条的渐近偏差与工作协方差阵无关, 而SUR核估计和光滑样条估计的渐近偏差却依赖于工作协方差阵. 本文主要研究了回归样条, 光滑样条和SUR核估计的效率问题. 通过模拟比较发现回归样条估计的表现比较稳定, 在大多数情况下比光滑样条估计和SUR核估计的效率高.     

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.     

Local influence in multilevel regression for growth curves

Influence analysis is important in modelling and identification of special patterns in the data. It is well established in ordinary regression. However, analogous diagnostics are generally not available for the multilevel regression model, in which estimation involves a complex iterative algorithm. This paper studies the local influence of small perturbations on the parameter estimates in the multilevel regression model with application to growth curves. The estimation is based on the iterative generalized least-squares (IGLS) method suggested by Goldstein (Biometrika 73 (1986) 43). The generalized influence function and generalized Cook statistic (Biometrika 84(1) (1997) 175) of IGLS of unknown parameters under some specific simultaneous perturbations are derived to study the joint influence of subject units on parameter estimators. The perturbation scheme is introduced through a variance–covariance matrix of error variables. A one-step approximation formula is suggested for simplifying the computations. The method is examined on growth-curve data.     

Variable selection for the partial linear single-index model

In this paper, we consider the issue of variable selection in partial linear single-index models under the assumption that the vector of regression coefficients is sparse. We apply penalized spline to estimate the nonparametric function and SCAD penalty to achieve sparse estimates of regression parameters in both the linear and single-index parts of the model. Under some mild conditions, it is shown that the penalized estimators have oracle property, in the sense that it is asymptotically normal with the same mean and covariance that they would have if zero coefficients are known in advance. Our model owns a least square representation, therefore standard least square programming algorithms can be implemented without extra programming efforts. In the meantime, parametric estimation, variable selection and nonparametric estimation can be realized in one step, which incredibly increases computational stability. The finite sample performance of the penalized estimators is evaluated through Monte Carlo studies and illustrated with a real data set.     

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.     

Quantile estimations via modified Cholesky decomposition for longitudinal single-index models

Quantile regression is a powerful complement to the usual mean regression and becomes increasingly popular due to its desirable properties. In longitudinal studies, it is necessary to consider the intra-subject correlation among repeated measures over time to improve the estimation efficiency. In this paper, we focus on longitudinal single-index models. Firstly, we apply the modified Cholesky decomposition to parameterize the intra-subject covariance matrix and develop a regression approach to estimate the parameters of the covariance matrix. Secondly, we propose efficient quantile estimating equations for the index coefficients and the link function based on the estimated covariance matrix. Since the proposed estimating equations include a discrete indicator function, we propose smoothed estimating equations for fast and accurate computation of the index coefficients, as well as their asymptotic covariances. Thirdly, we establish the asymptotic properties of the proposed estimators. Finally, simulation studies and a real data analysis have illustrated the efficiency of the proposed approach.

     

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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.     

Explicit estimators of parameters in the Growth Curve model with linearly structured covariance matrices

Estimation of parameters in the classical Growth Curve model, when the covariance matrix has some specific linear structure, is considered. In our examples maximum likelihood estimators cannot be obtained explicitly and must rely on optimization algorithms. Therefore explicit estimators are obtained as alternatives to the maximum likelihood estimators. From a discussion about residuals, a simple non-iterative estimation procedure is suggested which gives explicit and consistent estimators of both the mean and the linear structured covariance matrix.     

多元失效时间数据的一般边际半参数危险率模型研究

在生物医学研究中,多元失效时间数据非常常见.该文提出用一般边际半参数危险率回归模型来分析多元失效时间数据.此模型包括了三种常用边际模型:边际比例风险模型、边际加速失效时间模型和边际加速危险模型作为子模型.对于模型中的回归系数,可以通过估计方程的方法来估计它,同时也给出了基准累积危险率函数的估计.得到的估计可以证明是相合的和渐近正态的.     

Regression estimation with transformed auxiliary variates

The regression estimation of the mean of a primary survey variable and the estimation of the regression equation are considered in the finite population with transformed auxiliary variables. Large sample properties of estimators are developed. The effects of estimating auxiliary variates on estimators are investigated.     

Double Shrinkage Estimation of Common Coefficients in Two Regression Equations with Heteroscedasticity

The problem of estimating the common regression coefficients is addressed in this paper for two regression equations with possibly different error variances. The feasible generalized least squares (FGLS) estimators have been believed to be admissible within the class of unbiased estimators. It is, nevertheless, established that the FGLS estimators are inadmissible in light of minimizing the covariance matrices if the dimension of the common regression coefficients is greater than or equal to three. Double shrinkage unbiased estimators are proposed as possible candidates of improved procedures.     

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.     

Sequences of bias-adjusted covariance matrix estimators under heteroskedasticity of unknown form

The linear regression model is commonly used by practitioners to model the relationship between the variable of interest and a set of explanatory variables. The assumption that all error variances are the same, known as homoskedasticity, is oftentimes violated when cross sectional data are used. Consistent standard errors for the ordinary least squares estimators of the regression parameters can be computed following the approach proposed by White (Econometrica 48:817–838, 1980). Such standard errors, however, are considerably biased in samples of typical sizes. An improved covariance matrix estimator was proposed by Qian and Wang (J Stat Comput Simul 70:161–174, 2001). In this paper, we improve upon the Qian–Wang estimator by defining a sequence of bias-adjusted estimators with increasing accuracy. The numerical results show that the Qian–Wang estimator is typically much less biased than the estimator proposed by Halbert White and that our correction to the former can be quite effective in small samples. Finally, we show that the Qian–Wang estimator can be generalized into a broad class of heteroskedasticity-consistent covariance matrix estimators, and our results can be easily extended to such a class of estimators.     

MOMENT ESTIMATION FOR MULTIVARIATE EXTREME VALUE DISTRIBUTION

Moment estimation for multivariate extreme value distribution is described in this paper. Asymptotic covariance matrix of the estimators is given. The relative efficiencies of moment estimators as compared with the maximum likelihood and the stepwise estimators are computed. We show that when there is strong dependence between the variates, the generalized variance of moment estimators is much lower than the stepwise estimators. It becomes more obvious when the dimension increases.     

Partial functional linear quantile regression

This paper studies estimation in partial functional linear quantile regression in which the dependent variable is related to both a vector of finite length and a function-valued random variable as predictor variables. The slope function is estimated by the functional principal component basis. The asymptotic distribution of the estimator of the vector of slope parameters is derived and the global convergence rate of the quantile estimator of unknown slope function is established under suitable norm. It is showed that this rate is optimal in a minimax sense under some smoothness assumptions on the covariance kernel of the covariate and the slope function. The convergence rate of the mean squared prediction error for the proposed estimators is also be established. Finite sample properties of our procedures are studied through Monte Carlo simulations. A real data example about Berkeley growth data is used to illustrate our proposed methodology.     

Quasi-likelihood estimation of average treatment effects based on model information

In this paper, the estimation of average treatment effects is considered when we have the model information of the conditional mean and conditional variance for the responses given the covariates. The quasi-likelihood method adapted to treatment effects data is developed to estimate the parameters in the conditional mean and conditional variance models. Based on the model information, we define three estimators by imputation, regression and inverse probability weighted methods. All the estimators are shown asymptotically normal. Our simulation results show that by using the model information, the substantial efficiency gains are obtained which are comparable with the existing estimators.     

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.     

Estimation of means of multivariate normal populations with order restriction

Multivariate isotonic regression theory plays a key role in the field of statistical inference under order restriction for vector valued parameters. Two cases of estimating multivariate normal means under order restricted set are considered. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but are restricted by partial order. This paper shows that when covariance matrices are known, the estimator given by this paper always dominates unrestricted maximum likelihood estimator uniformly, and when covariance matrices are unknown, the plug-in estimator dominates unrestricted maximum likelihood estimator under the order restricted set of covariance matrices. The isotonic regression estimators in this paper are the generalizations of plug-in estimators in unitary case.     

Covariance operator estimation of a functional autoregressive process with random coefficients

We deal with the covariance and cross covariance operators estimation of a Hilbert space valued autoregressive process with random coefficients. We establish bounds for empirical estimators in mean square error and almost sure convergence in Hilbert–Schmidt norm. Consistent estimators of the eigenvalues are also derived.