Suppose that the random variables Y1, . . . , Yn satisfy

Yi = xi + i , i = 1, . . . , n,

where x1, . . . , xn are fixed constants, and 1, . . . , n are i.i.d. variables with standard normal distritbution N(0, 1).

(1) (10 points) Find the M.L.E., n (1) , of , and show that it is an unbiased estimator of .

(2) (10 points) Define another two estimators (2) n and (3) n as follows:

(2) n = Pn i=1 P Yi/ n i=1 xi , (3) n = 1 /n *Xn i=1 Yi/ xi .

Show that both of them are unbiased.

(3) (10 points) Find the relative efficiency of (1) n relative to (2) n and (3) n respectively, i.e., eff( (1) n , (2) n ) and eff( (1) n , (3) n ).

(4) (5 points) Use the M.L.E. to construct a two-sided (1 ) confidence interval for .