Equivariant estimators and a special group structure

Given a G-invariant family of distributions and under suitable hypotheses concerning G, we characterize the form of G-equivariant estimators. In fact, corresponding to each G-equivariant estimator is an appropriate G-invariant function and conversely.

In the course of characterizing the G-equivariant estimators, we obtain two maximal invariant functions. Some properties of these functions are obtained and in particular we calculate their densities with respect to an appropriate Haar measure.

Finally, we consider an invariant estimator problem, the problem of estimating the orbit of a parameter. It is seen that this invariant problem may be referred back to an equivariant one. A loss function for the invariant problem is defined in such a way that the minimumr risk invariant estimator corresponds to the minimumr risk equivariant estimator within a subclass of all equivariant estimators.