Robust estimators under a functional common principal components model

When dealing with several populations of functional data, equality of the covariance operators is often assumed even when seeking for a lower-dimensional approximation to the data. Usually, if this assumption does not hold, one estimates the covariance operator of each group separately, which leads to a large number of parameters. As in the multivariate setting, this is not satisfactory since the covariance operators may exhibit some common structure, as is, for instance, the assumption of common principal directions. The existing procedures to estimate the common directions are sensitive to atypical observations. For that reason, robust projection-pursuit estimators for the common directions under a common principal component model are considered. A numerical method to compute the first directions is also provided. Under mild conditions, consistency results are obtained. A Monte Carlo study is performed to compare the finite sample behaviour of the estimators based on robust scales and on the standard deviation. The usefulness of the proposed approach is illustrated on a real data set.