A Smooth Simultaneous Confidence Corridor for the Mean of Sparse Functional Data

Functional data analysis (FDA) has become an important area of statistics research in the recent decade, yet a smooth simultaneous confidence corridor (SCC) does not exist in the literature for the mean function of sparse functional data. SCC is a powerful tool for making statistical inference on an entire unknown function, nonetheless classic "Hungarian embedding" techniques for establishing asymptotic correctness of SCC completely fail for sparse functional data. We propose a local linear SCC and a shoal of confidence intervals (SCI) for the mean function of sparse functional data, and establish that it is asymptotically equivalent to the SCC of independent regression data, using new results from Gaussian process extreme value theory. The SCC procedure is examined in simulations for its superior theoretical accuracy and performance, and used to analyze growth curve data, confirming findings with quantified high significance levels. Supplementary materials for this article are available online.