ABSTRACT
In Chapter 4 we studied the interrelationships of L-estimators, L-functionals, and differentiable statistical functionals established their asymptotic representations from some basic results. M-estimators are also expressible as statistical functionals but in an implicit manner. This in turn requires a more elaborate treatment. The approach based on the Hadamard (or compact) differentiable functionals requires bounded score functions as well as other regularity conditions. The bounded condition is often justified for robust functionals, but excludes the maximum likelihood estimators, the precursors of M-estimators. thus, we shall concentrate on alternative methods, based mostly on the uniform asymptotic linearity of M-statistics in the associated parameter(s); this method constitutes the main theme of the current chapter. A variety of onestep and studentized versions of M-estimators of location/regression parameters will be considered and their related first-order and second-order asymptotic representations will be systematically presented.
