  This article introduces trimmed estimators for the mean and covariance
function of general functional data. The estimators are based on a new measure
of outlyingness or data depth that is well defined on any metric space,
although this paper focuses on Euclidean spaces. We compute the breakdown point
of the estimators and show that the optimal breakdown point is attainable for
the appropriate choice of tuning parameters. The small-sample behavior of the
estimators is studied by simulation, and we show that they have better
outlier-resistance properties than alternative estimators. This is confirmed by
two real-data applications, that also show that the outlyingness measure can be
used as a graphical outlier-detection tool in functional spaces where visual
screening of the data is difficult.
