Robust estimators and probability integral transformations

This article gives a construction of a weighted mean which has good robust properties (qualitative robustness, bounded influence function, high breakdown points). The construction is based on M-functionals with smooth defining functions which are used to control weighting. This construction has the advantage of the computational speed and opens up the possibility of allowing the weighting to take the shape of the original data set into account. This method can be applied to some other statistical problems: the estimation of shape parameter at the Student- and Weibull-family, examination for sufficient statistics of Ornstein-Uhlenbeck process.