1.(a) Show that the UNIF(0, ) density is a scale invariant family.

(b) Show that the U.M.V.U.E. of based on a random sample X1, . . . , Xn isa scale equivariant estimator

2.Suppose Yij, i = 1, 2; j = 1, . . . , n are independent random variables with E(Yij) = + i and V ar(Yij) = 2 where 1 + 2 = 0. (a) Find the best linear unbiased estimator of 1.

3.Suppose X1, . . . ,Xn is a random sample from the Bernoulli() distribution. Suppose also that (i, . . . , n) are independent N(0, 2) random variables independent of the Xis. Define Yi = Xi + i, i = 1, . . . , n. We observe only the values (Xi, Yi), i = 1, . . . , n. The parameter is unknown and the is are unobserved. Define the estimating function [; (X, Y )] = Pn i=1 (Yi Xi). (a) Show that this is an unbiased estimating function for . (b) Find the estimator which satisfies [; (X, Y )] = 0. Is an unbiased estimator of ? (c) Construct an approximate 95% C.I. for .