When collections of functional data are too large to be exhaustively
observed, survey sampling techniques provide an effective way to estimate
global quantities such as the population mean function. Assuming functional
data are collected from a finite population according to a probabilistic
sampling scheme, with the measurements being discrete in time and noisy, we
propose to first smooth the sampled trajectories with local polynomials and
then estimate the mean function with a Horvitz-Thompson estimator. Under mild
conditions on the population size, observation times, regularity of the
trajectories, sampling scheme, and smoothing bandwidth, we prove a Central
Limit theorem in the space of continuous functions. We also establish the
uniform consistency of a covariance function estimator and apply the former
results to build confidence bands for the mean function. The bands attain
nominal coverage and are obtained through Gaussian process simulations
conditional on the estimated covariance function. To select the bandwidth, we
propose a cross-validation method that accounts for the sampling weights. A
simulation study assesses the performance of our approach and highlights the
influence of the sampling scheme and bandwidth choice.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ443 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
