Pitman closest equivariant estimators and predictors under location-scale models

For location, scale and location-scale models, which are common in practical applications, we derive optimum equivariant estimators and predictors using the Pitman closeness criterion. This approach is very robust with respect to the choice of the loss function as it only requires the loss function to be strictly monotone. We also prove that, in general, the Pitman closeness comparison of any two equivariant predictors depends on the unknown parameter only through a maximal invariant, and hence it is independent of the parameter when the parameter space is transitive. We present several examples illustrating applications of our theoretical results.