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Estimation of the mean and covariance functions is very important to analyze multivariate longitudinal and sparse functional data. We define a new covariance function that not only consider the correlation of different observed responses for the same biomarker but different biomarkers. Full quasi-likelihood and the kernel method are used to approximate mean and covariance functions, the covariance decomposition is considered to decompose covariance functions to correlation function and variance function. We use the full quasi-likelihood to solve measurement errors variance  λ  and choose the iterative algorithm to update the multivariate mean and covariance functions until convergence. Gaussian kernel and leave-one-out cross-validation are used to select bandwidth  h . Finally, we give theoretical properties of the unknown functions and prove their convergence. Simulation and application results show the effectiveness of our proposed method.

Estimation of the nonparametric mean and covariance functions for multivariate longitudinal and sparse functional data