Let X1, X2, . . . be i.i.d. variables uniformly distributed on (, 2), and > 0 is the unknown parameter. Define Gn and Hn as follows: Gn = (X1 X2 Xn) ^1/n , Hn = n X 1^-1 + + X n^1 .

(1) (15 points) Find the constant c1 and c2 such that (1) n = c1 Gn and (2) n = c2 Hn are both consistent estimators of .

(2) (10 points) Find the maximum likelihood estimator (3) n of and show that the M.L.E. is a consistent estimator of .