Estimation Theories For Mean-covariance Models With Longitudinal Data

Longitudinal data arises frequently in the epidemiology,biological research.economics and social sciences,where subjects are measured repeatedly over time Thus the observations on the same subject are intrinsically correlated.Working structures are commonly used for the within-subject covariance matrix.Howev-er,the misspecification of the working covariance structure may result in a great loss of estimation efficiency of the mean parameters.Furthermore,the covariance matrix itself may be of scientific interest,but suffers from the positive-definiteness constraint and the high-dimensionality of the number of parameters.In addi-tion,multivariate longitudinal data always suffers from much more complicated covariance structures.Nowadays,one of the most popular approaches dealing with the covariance structure is to establish regression models based on the Cholesky decomposition for the covariance structure.However,the existing approaches mainly focus on the modified Cholesky decomposition with univariate longitudinal data.The other Cholesky decompositions need to be further investigated,and more complicated multivariate longitudinal data deserves to be considered.In this dissertation,we focus on modeling the covariance structure in the generalized estimating equations based on the autoregressive moving average Cholesky decomposition or alternative Cholesky decomposition,developing the robust estimation approach with respect to outliers or heavy tailed distribution based on the modified Cholesky decom-position and exponential squared loss function with univariate longitudinal data,and establishing new Cholesky decompositions with multivariate longitudinal da-ta.More specifically,the main research contents of this dissertation are described as follows.We begin with developing generalized estimating equations for the regres-sion parameters in joint mean-covariance models with balanced or unbalanced longitudinal data,motivated by the autoregressive moving average Cholesky de-composition or alternative Cholesky decomposition.Firstly,the autoregressive moving average Cholesky decomposition is obtained by combining the autoregres-sive Cholesky decomposition and moving average Cholesky decomposition,and thus able to parameterize more general covariance structures.The entries in this decomposition have reasonable statistical interpretation,and the positive definite-ness constraint of the covariance matrix can be automatically satisfied via this decomposition.Then these entries can be modeled by regression models,and the regression parameters can be computed by the quasi Fisher iterative algorithm The parameter estimators both in mean and covariance models are demonstrated to be consistent and asymptotically normal.Simulations and real data analysis are carried out to illustrate the proposed approach.Secondly,the alternative C-holesky decomposition is closely related to the moving average representation of"standardized",repeated measurements on a subject,which causes that estimation of the correlation matrix is robust against misspecification of the model for inno-vation variances.We then establish the generalized estimating equations,develop the computing algorithm,demonstrate the asymptotic properties of the parame-ter estimators both in mean and covariance models,and carry out some numerical studies to investigate the performance of the proposed approachWhen longitudinal data contains outliers,the classical least squares approach is known to be not robust.To solve this issue,the exponential squared loss function with a tuning parameter has been investigated for longitudinal data.However,to our knowledge,there is no paper to investigate the robust estimation procedure against outliers within the framework of mean-covariance regression analysis for longitudinal data using the exponential squared loss function.Firstly,we propose a robust estimation approach for the model parameters of the mean and gener-alized autoregressive parameters with longitudinal data based on the exponential squared loss function.The proposed estimators can be shown to be asymptotically normal under certain conditions.Moreover,we develop an iteratively reweighted least squares algorithm to calculate the parameter estimates,and the balance be-tween the robustness and efficiency can be achieved by choosing appropriate data adaptive tuning parameters.Simulation studies and real data analysis are carried out to illustrate the finite sample performance of the proposed approach.Sec-ondly,we generalize the mean model from linear to partially nonlinear.Then we further develop the robust estimation approach and a minorization-maximization algorithm to calculate the parameter estimates,and carry out some numerical studies to evaluate the performance of the proposed approachMultivariate longitudinal data is often encountered in the jobs of statisticians and practitioners.It is challenging to model the covariance matrix due to the complex structure of correlations among multiple responses.For this modeling task,several effective Cholesky decomposition based methods have been stud-ied.However,direct interpretation of the covariation structure among multiple responses is still less well investigated to the best of our knowledge.We propose a joint mean-variance correlation modeling method based on the triangular angles parameterization for the correlation matrix of bivariate longitudinal data.The proposed unconstrained parameterization is able to automatically eliminate the positive definiteness constraint of the correlation matrix and leads to the afore-mentioned direct interpretation.Furthermore,the standard deviation matrix is diagonal rather than block-diagonal,so the positive-definiteness constraint of this matrix can be easily satisfied.The entries of the proposed decomposition are mod-eled by regression models,and the maximum likelihood estimators of regression parameters in joint models are obtained.The resulting estimators are shown to be consistent and asymptotically normal.Simulations and a study of poplar growth illustrate that the proposed method performs wellThere are little methods focusing on the robustness of estimating the correla-tion matrix of multivariate longitudinal data.We propose an alternative Cholesky block decomposition for the covariance matrix of multivariate longitudinal data The new unconstrained parameterization is capable to automatically eliminate the positive definiteness constraint of the covariance matrix and robustly estimate the correlation matrix with respect to the model misspecification of the nested innovation variance matrices.Note that the new unconstrained parameters pos-sess reasonable statistical interpretation,then these parameters are modeled by regression models,and the maximum likelihood estimators of the regression pa-rameters in joint mean-covariance models are computed by a quasi Fisher iterative algorithm.The resulting estimators are shown to be consistent and asymptotically normal.Simulations and real data analysis illustrate the finite sample performance of the new method.