Semiparametric Estimation of Covariance Matrixes for Longitudinal Data

• Published in: Journal of the American Statistical Association Vol. 103; no. 484; pp. 1520 - 1533
• Main Authors: Fan, Jianqing, Wu, Yichao
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.12.2008; American Statistical Association; Taylor & Francis Ltd
• Subjects: Applications; Asymptotic methods; Asymptotic properties; Context; Correlation analysis; Correlation structure; Correlations; Covariance; Covariance matrices; Covariance matrix; Data points; Difference-based estimation; Estimation bias; Estimation methods; Estimators; Exact sciences and technology; General topics; Linear inference, regression; Linear models; Longitudinal data; Mathematics; Matrix; Maximum likelihood estimators; Mean; Multivariate analysis; Normality; Parameter estimation; Parameters; Parametric models; Probability and statistics; Quasi-maximum likelihood; Regression analysis; Regression models; Robustness (mathematics); Sciences and techniques of general use; Simulation; Standard deviation; Statistical methods; Statistical variance; Statistics; Theory and Methods; Variance analysis; Varying-coefficient partially linear model; Correlation; Conditional distribution; Variance function; Statistical distribution; Variance estimation; Estimator robustness; Non parametric estimation; Multivariate analysis; Statistical simulation; Asymptotic convergence; Linear model; Variance; Semiparametric model; Covariance; Difference-based estimation; Regression function; Correlation function; Data structure; Covariance analysis; Linear regression; Asymptotic behavior; Statistical association; Covariance matrix; Semiparametric method; Variance analysis; Mean estimation; Statistical method; Statistical regression; Regression coefficient; Correlation analysis; Regression model; Asymptotic normality; Quasi-maximum likelihood: Varying-coefficient partially linear model; Quasi likelihood; Correlation structure; Maximum likelihood; Application; Biased estimation
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/016214508000000742

Estimation of longitudinal data covariance structure poses significant challenges because the data usually are collected at irregular time points. A viable semiparametric model for covariance matrixes has been proposed that allows one to estimate the variance function nonparametrically and to estimate the correlation function parametrically by aggregating information from irregular and sparse data points within each subject. But the asymptotic properties of the quasi-maximum likelihood estimator (QMLE) of parameters in the covariance model are largely unknown. We address this problem in the context of more general models for the conditional mean function, including parametric, nonparametric, or semiparametric. We also consider the possibility of rough mean regression function and introduce the difference-based method to reduce biases in the context of varying-coefficient partially linear mean regression models. This provides a more robust estimator of the covariance function under a wider range of situations. Under some technical conditions, consistency and asymptotic normality are obtained for the QMLE of the parameters in the correlation function. Simulation studies and a real data example are used to illustrate the proposed approach.