Depth measures, Functional data, Projections method, Supervised classification, Primary 62G07, Secondary 62G20,

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SUMMARYThis paper analyzes outlier detection for functional data by means of functional depths, which measures the centrality of a given curve within a group of trajectories providing center-outward orderings of the set of curves. We give some insights of the usefulness of looking for outliers in functional datasets and propose a method based in depths for the functional outlier detection. The performance of the proposed procedure is analyzed by several Monte Carlo experiments. Finally, we illustrate the procedure by finding outliers in a dataset of NO x (nitrogen oxides) emissions taken from a control station near an industrial area.

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Acknowledgements: We are very grateful to Kazaros Kazarian for some useful conversations. The useful and constructive comments of three referees are gratefully acknowledged. This work has been partially supported by DGESIC Spanish Grants PB97-0021 (A. Cuevas) and PB98-0182-C02-02, PGIDT99MA 20701 (M. Febrero). The problem of simple linear regression, with functional explanatory variables and functional response, is considered under a fixed design model. An estimator for the underlying linear operator is proposed. Its consistency is proved under some conditions which ensure that the design is informative enough. The classical calibration (or "inverse regression") problem is considered and a consistent estimator is analyzed. A simulation study is also given. The proposed method is computationally feasible and is not hard to implement in practice.