Parametric estimation with a class of M-estimators

Consider a class of M-estimators indexed by a criterion function ψ. When the function ψ is taken to be in a class of functions $ \mathcal{F} $ \mathcal{F} , a family of processes indexed by the class $ \mathcal{F} $ \mathcal{F} is obtained and called M-processes. Pooling the M-estimators in such class may be used to define new kind of estimators. In order to get the asymptotic properties of these pooled estimators, the convergence in probability of the corresponding M-process is studied uniformly on $ \mathcal{F} $ \mathcal{F} together with their weak convergence towards a Gaussian process. An application to location estimation is presented and discussed.