  Irregular functional data in which densely sampled curves are observed over
different ranges pose a challenge for modeling and inference, and sensitivity
to outlier curves is a concern in applications. Motivated by applications in
quantitative ultrasound signal analysis, this paper investigates a class of
robust M-estimators for partially observed functional data including functional
location and quantile estimators. Consistency of the estimators is established
under general conditions on the partial observation process. Under smoothness
conditions on the class of M-estimators, asymptotic Gaussian process
approximations are established and used for large sample inference. The large
sample approximations justify a bootstrap approximation for robust inferences
about the functional response process. The performance is demonstrated in
simulations and in the analysis of irregular functional data from quantitative
ultrasound analysis.
