Assume that X has a density of the form and that it is desired to estimate under a strictly convex loss which depends only on Prove that is a best invariant estimator. Assume that X has a...

Assume that X has a density of the form

and that it is desired to estimate

under a strictly convex loss which depends only on

Prove that

is a best invariant estimator.

Assume that X has a unimodal density of the form

(so that f( z) is non-increasing for

and that it is desired to estimate

under a loss of the form

where W(z) is non-decreasing for

Prove that

is a best invariant estimator.