  New estimators for the mean and the covariance function for partially
observed functional data are proposed using a detour via the fundamental
theorem of calculus. The new estimators allow for a consistent estimation of
the mean and covariance function under specific violations of the
missing-completely-at-random assumption. The requirements of the estimation
procedure can be tested using a sequential multiple hypothesis test procedure.
An extensive simulation study compares the new estimators with the classical
estimators from the literature in different missing data scenarios. The
proposed methodology is motivated by the practical problem of estimating the
mean price curve in the German Control Reserve Market. In this auction market,
price curves are only partially observable and the underlying missing data
mechanism depends on systematic trading strategies which clearly violate the
missing-completely-at-random assumption. In contrast to the classical
estimators, the new estimators lead to useful estimates of the mean and
covariance functions. Supplementary materials are provided online.
