Inferential procedures for partially observed functional data

In functional data analysis it is usually assumed that all functions are completely, densely or sparsely observed on the same domain. Recent applications have brought attention to situations where each functional variable may be observed only on a subset of the domain while no information about the function is available on the complement. Various advanced methods for such partially observed functional data have already been developed but, interestingly, some essential methods, such as K-sample tests of equal means or covariances and confidence intervals for eigenvalues and eigenfunctions, are lacking. Without requiring any complete curves in the data, we derive asymptotic distributions of estimators of the mean function, covariance operator and eigenelements and construct hypothesis tests and confidence intervals. To overcome practical difficulties with storing large objects in computer memory, which arise due to partial observation, we use the nonparametric bootstrap approach. The proposed methods are investigated theoretically, in simulations and on a fragmentary functional data set from medical research.