5 Conclusion and perspectives for research
Hervé Cardot, Alain Dessertaine, Camelia Goga, Étienne Josserand and Pauline Lardin
In this study, we have implemented and compared different strategies for using auxiliary information for estimating, and constructing confidence bands for, the mean of data in the form of curves. This information can be taken into consideration at the time of sampling by using unequal probability designs or during estimation with simple random sampling without replacement, assisted by a functional-response regression model. It seems clear from our example of electricity consumption curves that when total consumption for the previous week is known, the precision of estimators of the mean can be greatly improved compared with an SRSWOR-type sampling.
Also, in this context of large samples and high-dimensional data, it also seems possible to construct, for these different strategies, confidence bands that have empirical coverage rates close to the desired rates. The two considered approaches--estimation of the covariance function and simulation of Gaussian or bootstrap processes-seem to perform comparably in terms of the width of the confidence bands; the main difference is in the computation time. The bootstrap, which seems more general because it does not require having a good estimator of the covariance function, proves to be much slower in practice.
Sometimes, in these flows of large-scale data, there are losses of information owing to signal transmission problems. The end result is that the utility has incomplete records of some trajectories. This issue, of partial non-response, can probably be dealt with by considering adaptations of classical non-response techniques (Haziza 2009) in the functional context. A fundamental question, then, is how to construct good estimators of the covariance function.
Acknowledgements
We wish to thank the anonymous referees as well as Guillaume Chauvet and Jean-Claude Deville for their helpful comments, which led to improvements in this study.
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