. 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 *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 .

Problem 3 (35 points). Suppose that the random variables Y1, ..., Y, satisfy Yi = Britei, i = 1, . ..,n, where r1, . .., In are fixed constants, and #1, ..., En are i.i.d. variables with standard normal distritbution N(0, 1). (1) (10 points) Find the M.L.E., S,", of B, and show that it is an unbiased estimator of B. (2) (10 points) Define another two estimators On and ," as follows: B(2) = B(3) = Yi n i= 1 Show that both of them are unbiased. (3) (10 points) Find the relative efficiency of An relative to , and , respectively, i.e., eff( 3,"), B,") and eff( B, ), B,3)). (4) (5 points) Use the M.L.E. to construct a two-sided (1 - o) confidence interval for B