From sparse to dense functional data: phase transitions from a simultaneous inference perspective

Leheng Cai, Qirui Hu

Abstract: We aim to develop unified tools for simultaneous inference regarding the mean function of functional data, covering situations from sparse to dense observations. We first establish a unified Gaussian approximation applicable across arbitrary sampling schemes, which facilitates the construction of simultaneous confidence bands for mean functions using the B-spline estimator. Subsequently, we explore the conditions leading to phase transitions by decomposing the asymptotic variance of the approximating Gaussian processes. These conditions are determined by the relationship between the number of knots, the sample sizes, and the number of observations per trajectory. As an extension, we also consider the orthogonal series estimator, highlighting how phase transitions are affected by the sup-norm, the approximation power, and other features of the employed orthonormal basis. Our extensive simulations strongly support our theoretical findings and further illustrate the variability of the asymptotic distribution via the decomposition of asymptotic variance we obtain. The developed method is further applied to body fat data and traffic data.

External IDs:dblp:journals/sac/CaiH25