On multivariate higher order von Mises expansions

In this paper we review the von Mises calculations of higher order for statistical functionals of one variable. For the functionals of several variables, the higher order Gateaux differentials are defined leading to the corresponding multivariate higher order von Mises expansions. These expansions are used to analyze the bias of the corresponding statistical functionals. The second and third order derivatives are computed for M-estimates. Applications of these expansions to study the bias of M-estimates of location and to simultaneous M-estimates of location and scale are also given. Received: July 2000