Abstract: 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 study investigates a class of robust M-estimators for partially observed functional data, including functional location and quantile estimators. The 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 using a bootstrap approximation for robust inferences about the functional response process. The performance of the proposed estimators is demonstrated by means of simulations and an analysis of irregular functional data from quantitative ultrasound analysis.

Key words and phrases: Bootstrap, functional central limit theorem, functional quantile, L²-norm test, trend analysis.