5. Let X U(,2). (a) Show that (X(1),X(n)) is jointly sufficient statistic for . (b) Verify...

5. Let X ∼ U(θ,2θ).

(a) Show that (X(1),X(n)) is jointly sufficient statistic for θ.

(b) Verify whether or not (X(n)

− X(1)) is an unbiased estimator of θ. Find an ancillary statistic.

(c) D etermine the best invariant estimator of θ under the loss function L(θ,a) =

1− a

θ

2.

6. Let fθ(x) =

1 2

exp{−|x−θ|}.

Find the Pitman estimator of θ.