Framelet block thresholding estimator for sparse functional data

Nonparametric estimation of mean and covariance functions based on discretely observed data is important in functional data analysis. In this paper, we propose a framelet block thresholding method for the case of sparsely observed functional data. The procedure is easily implemented and the resultant estimators are represented as explicit B-spline expressions. For sparsely observed functional data, we establish, under some mild conditions but without knowing the smoothness parameter, convergence rates of mean integrated squared errors for mean and covariance estimators respectively. In particular, the mean estimator attains minimax optimal rate. The simulated and real data examples are provided to offer empirical support of the theoretical properties. Compared to the existing methods, the proposed method outperforms in adapting automatically to local variations.