Estimation of Large Covariance Matrices of Longitudinal Data with Basis Function Approximations

The major difficulties in estimating a large covariance matrix are the high dimensionality and the positive definiteness constraint. To overcome these difficulties, we propose to apply smoothing-based regularization and utilize the modified Cholesky decomposition of the covariance matrix. In our proposal, the covariance matrix is diagonalized by a lower triangular matrix, whose subdiagonals are treated as smooth functions. These functions are approximated by splines and estimated by maximizing the normal likelihood. In our framework, the mean and the covariance of the longitudinal data can be modeled simultaneously and missing data can be handled in a natural way using the EM algorithm. We illustrate the proposed method via simulation and applying it to two real data examples, which involve estimation of 11 by 11 and 102 by 102 covariance matrices.