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When dealing with very large datasets of functional data, survey samplingapproaches are useful in order to obtain estimators of simple functionalquantities, without being obliged to store all the data. We propose here aHorvitz--Thompson estimator of the mean trajectory. In the context of asuperpopulation framework, we prove under mild regularity conditions that weobtain uniformly consistent estimators of the mean function and of its variancefunction. With additional assumptions on the sampling design we state afunctional Central Limit Theorem and deduce asymptotic confidence bands.Stratified sampling is studied in detail, and we also obtain a functionalversion of the usual optimal allocation rule considering a mean variancecriterion. These techniques are illustrated by means of a test population ofN=18902 electricity meters for which we have individual electricity consumptionmeasures every 30 minutes over one week. We show that stratification cansubstantially improve both the accuracy of the estimators and reduce the widthof the global confidence bands compared to simple random sampling withoutreplacement.

Horvitz-Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling