Prediction intervals and bands with improved coverage for functional data under noisy discrete observation

We revisit the classic situation in functional data analysis in which curves are observed at discrete, possibly sparse and irregular, arguments with observation noise. We focus on the reconstruction of individual curves by prediction intervals and bands. The standard approach consists of two steps: first, one estimates the mean and covariance function of curves and observation noise variance function by, e.g. penalized splines, and second, under Gaussian assumptions, one derives the conditional distribution of a curve given observed data and constructs prediction sets with required properties, usually employing sampling from the predictive distribution. This approach is well established, commonly used and theoretically valid but practically, it surprisingly fails in its key property: prediction sets constructed this way often do not have the required coverage. The actual coverage is lower than the nominal one. We investigate the cause of this issue and propose a computationally feasible remedy that leads to prediction regions with a much better coverage. Our method accounts for the uncertainty of the predictive model by sampling from the approximate distribution of its spline estimators whose covariance is estimated by a novel sandwich estimator. Our approach also applies to the important case of covariate-adjusted models.