Tolerance bands for functional data

Often the object of inference in biomedical applications is a range that brackets a given fraction of individual observations in a population. A classical estimate of this range for univariate measurements is a tolerance interval. This article develops its natural extension for functional measurements, a tolerance band, and proposes a methodology for constructing its pointwise and simultaneous versions that incorporates both sparse and dense functional data. Assuming that the measurements are observed with noise, the methodology uses functional principal component analysis in a mixed model framework to represent the measurements and employs bootstrapping to approximate the tolerance factors needed for the bands. The proposed bands also account for uncertainty in the principal components decomposition. Simulations show that the methodology has, generally, acceptable performance unless the data are quite sparse and unbalanced, in which case the bands may be somewhat liberal. The methodology is illustrated using two real datasets, a sparse dataset involving CD4 cell counts and a dense dataset involving core body temperatures.