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When collections of functional data are too large to be exhaustivelyobserved, survey sampling techniques provide an effective way to estimateglobal quantities such as the population mean function. Assuming functionaldata are collected from a finite population according to a probabilisticsampling scheme, with the measurements being discrete in time and noisy, wepropose to first smooth the sampled trajectories with local polynomials andthen estimate the mean function with a Horvitz-Thompson estimator. Under mildconditions on the population size, observation times, regularity of thetrajectories, sampling scheme, and smoothing bandwidth, we prove a CentralLimit theorem in the space of continuous functions. We also establish theuniform consistency of a covariance function estimator and apply the formerresults to build confidence bands for the mean function. The bands attainnominal coverage and are obtained through Gaussian process simulationsconditional on the estimated covariance function. To select the bandwidth, wepropose a cross-validation method that accounts for the sampling weights. Asimulation study assesses the performance of our approach and highlights theinfluence of the sampling scheme and bandwidth choice.

Confidence bands for Horvitz-Thompson estimators using sampled noisy functional data