Partially Observed Functional Data: The Case of Systematically Missing Parts

By using a detour via the fundamental theorem of calculus, we propose a new estimation procedure that allows the consistent estimation of the mean and covariance function from partially observed functional data under specific violations of the missing-completely-at-random assumption. The requirements of our estimation procedure can be tested in practice using a sequential multiple hypothesis test procedure. We perform an extensive simulation study where we compare our estimators with the classical estimators from the literature in different missing data scenarios. The methodology proposed is motivated by the practical problem of estimating mean price curves in the German Control Reserve Market. In this auction market, price curves are only partially observable and the underlying missing data mechanism strongly depends on systematic trading strategies that clearly violate the missing-completely-at-random assumption. In contrast to the classical estimators from the literature, our estimators lead to useful estimates of the mean and covariance functions.