Rate accelerated inference for integrals of multivariate random functions

The computation of integrals is a fundamental task in the analysis of functional data, where the data are typically considered as random elements in a space of squared integrable functions. Effective unbiased estimation and inference procedures are proposed for integrals of uni- and multivariate random functions. Applications to key problems in functional data analysis involving random design points are examined and illustrated. In the absence of noise, the proposed estimates converge faster than the sample mean and standard numerical integration algorithms. The estimator also supports effective inference by generally providing better coverage with shorter confidence and prediction intervals in both noisy and noiseless settings.