Confidence Bands for Horvitz-Thompson Estimators of Sampled Noisy Functional Data
Hervé Cardot1, David Degras2, Etienne Josserand1
1Institut de Mathématiques de Bourgogne; 2SAMSI

The development of automated sensors gives now access to very large potential databases of signals sampled at very fine time scales. Major investments would then be required to transmit, store and analyze all this information. For such large samples of data that can be seen as functional data, survey sampling techniques are interesting alternatives compared to signal compression techniques when the target is a global indicator such as the mean temporal signal.

Survey sampling techniques, which consist in selecting randomly a part of a population with a controlled probabilistic selection process, are interesting candidates for estimating global quantities such as the mean when one has potential access to very large databases of functional data. Assuming the data are observed at some time instants and corrupted by noise, which may not be i.i.d., we propose to first smooth the individual discretized trajectories with local polynomials and then estimate the mean function with an Horvitz-Thompson estimator. We prove, under mild conditions on the sampling design, the regularity of the trajectories, the number of discretization points and the asymptotic behavior of the bandwidth a Central Limit Theorem in the space of continuous functions. We also state, under reasonable conditions on the bandwidth that the covariance function can be estimated consistently. It is then possible to build consistent global confidence bands by performing Gaussian process simulations, given the estimated covariance function. An illustration on a simulation study allows to highlight the role of the bandwidth and assess the performances of this approach.

Keywords: Central Limit Theorem; Supremum of Gaussian processes; Survey sampling; Correlated noise

Biography: Etienne Josserand is a Ph.D. student in the University of Burgundy in France.