Una definizione di funzione di influenza generalizzata nella teoria della robustezza statistica

The idea of influence function is recent (Hampel, 1974) but nowadays it is a well established statistical tool and it is crucial for the important role in robustness and stability analysis (Hampel, Ronchetti, Rousseeuw and Stahel, 1986). In this paper we introduce a generalized definition of the influence function and we prove a generalized Von Mises theorem. The robustness analysis using influence functions is strongly based on the existence of the derivative of a functional. The aim of this work is to propose a new approach which can be used when the derivative doesn’t exist. This new definition of influence function is based on a notion of first order generalized derivative. The paper is structured as follows. In the first section we recall the classical definition of influence function and a notion of measure on which it is based the definition of B-robustness. In the second section we will recall some mathematical tools which will be useful for the extensions of the third section.