From dense to sparse design: Optimal rates under the supremum norm for estimating the mean function in functional data analysis

In the setting of functional data analysis, we derive optimal rates of convergence in the supremum norm for estimating the H\"older-smooth mean function of a stochastic processes which is repeatedly and discretely observed at fixed, multivariate, synchronous design points and with additional errors. Similarly to the rates in $L_2$ obtained in Cai and Yuan (2011), for sparse design a discretization term dominates, while in the dense case the $\sqrt n$ rate can be achieved as if the $n$ processes were continuously observed without errors. However, our analysis differs in several respects from Cai and Yuan (2011). First, we do not assume that the paths of the processes are as smooth as the mean, but still obtain the $\sqrt n$ rate of convergence without additional logarithmic factors in the dense setting. Second, we show that in the supremum norm, there is an intermediate regime between the sparse and dense cases dominated by the contribution of the observation errors. Third, and in contrast to the analysis in $L_2$, interpolation estimators turn out to be sub-optimal in $L_\infty$ in the dense setting, which explains their poor empirical performance. We also obtain a central limit theorem in the supremum norm and discuss the selection of the bandwidth. Simulations and real data applications illustrate the results.