Covariance function versus covariance matrix estimation in efficient semi-parametric regression for longitudinal data analysis

Improving estimation efficiency for regression coefficients is an important issue in the analysis of longitudinal data, which involves estimating the covariance matrix of the within-subject errors. In the balanced or nearly balanced setting, we can also regard the covariance matrix of the dependent errors as the bivariate covariance function evaluated at specific time points. In this paper, we compare the performance of the proposed regularized-covariance-function-based estimator and the conventional high-dimensional covariance matrix estimator of the within-subject errors. It shows that when the number p of the time points in each subject is large enough compared to the number n of the subjects, i.e., p≫n1/4logn, the estimation errors of the high-dimensional covariance matrix will be accumulated, therefore the error bound of the proposed regularized-covariance-function-based estimator will be smaller than that of the high-dimensional covariance matrix estimator in Frobenius norm. We also assess the performance of these two estimators for the incomplete longitudinal data. All the comparisons and theoretical results are illustrated using both simulated and real data.