Consider x an observation from a C(θ,1) distribution. Under quadratic loss, show that all equivariant estimators have infinite risk. Propose a finite risk estimator other than the constant estimator.
(Berger (1985a)) Consider 
where 
is unknown. The hypothesis 
s to be tested against H1: f = f1 under the 0 − 1 loss.
a. Show that 
is a maximal invariant statistic for the group of transformations 
b. Deduce that every invariant test only depends on 
and that the optimal tests have the following rejection region:
where 
is the density of y under Hi.
