Robust joint mean-covariance model selection and time-varying correlation structure estimation for dependent data

Abstract

In longitudinal and spatio-temporal data analysis, repeated measurements from a subject can be either regional- or temporal-dependent. The correct specification of the within-subject covariance matrix cultivates an efficient estimation for mean regression coefficients.

In this thesis, robust estimation for the mean and covariance jointly for the regression model of longitudinal data within the framework of generalized estimating equations (GEE) is developed. 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.

Robust variable selection method in a joint mean and covariance model is considered, by proposing a set of penalized robust generalized estimating equations to estimate simultaneously the mean regression coefficients, the generalized autoregressive coefficients and innovation variances introduced by the modified Cholesky decomposition. The set of estimating equations select important covariate variables in both mean and covariance models together with the estimating procedure. Under some regularity conditions, the oracle property of the proposed robust variable selection method is developed. For these two robust joint mean and covariance models, simulation studies and a hormone data set analysis are carried out to assess and illustrate the small sample performance, which show that the proposed methods perform favorably by combining the robustifying and penalized estimating techniques together in the joint mean and covariance model.

Capturing dynamic change of time-varying correlation structure is both interesting and scientifically important in spatio-temporal data analysis. The time-varying empirical estimator of the spatial correlation matrix is approximated by groups of selected basis matrices which represent substructures of the correlation matrix. After projecting the correlation structure matrix onto the space spanned by basis matrices, varying-coefficient model selection and estimation for signals associated with relevant basis matrices are incorporated. The unique feature of the proposed model and estimation is that time-dependent local region signals can be detected by the proposed penalized objective function. In theory, model selection consistency on detecting local signals is provided. The proposed method is illustrated through simulation studies and a functional magnetic resonance imaging (fMRI) data set from an attention deficit hyperactivity disorder (ADHD) study.