Optimal estimation and goodness-of-fit testing of the mean for sparse longitudinal functional data

Abstract: We study the mean function of longitudinal functional data, where each subject contributes a small number of complete profiles over a general domain, observed at random visit times. The mean is projected onto an orthonormal basis in the time direction, and each coefficient function is estimated by a weighted average of the profiles. We consider deterministic and random weights, including closed-form, data-driven weights that dispense with the design density entirely. For each weighting scheme we derive non-asymptotic bounds on the integrated quadratic risk and an explicit optimal truncation level. All schemes share the same convergence rate, which we show to be minimax optimal over the corresponding coefficient-decay class, only the leading constants differ. We also provide a goodness-of-fit test for the mean function, and derive non-asymptotic bounds for the Gaussian approximation of its distribution under both the null and alternative hypotheses. Our estimation and testing procedures are easy to implement, fast, and perform well in applications.