Let(X1,...,Xn) be a random sample of random variables from a populationPwith E(Xi2)< andXbe the sample mean. Consider the estimation of=E(Xi).

(i) LetTn=X+/n , where n is a random variable satisfying n=0 with probability (1-1/n) and n=n3/2 with probability 1/n. Show that the bias of Tn is not the same as its asymptomtic bias for any distribution of P.

(ii) LetTn=X+/n where now n is a random variable that is independent of X1,...,Xn and n=0 with probability (1-2/n), n=n with probability 1/n and n=n also with probability 1/n. Show that the approximate mean squared error ofTn , the

approximate mean squared error ofX, and the mean squared error ofX are the same,but the mean squared error of Tn is larger than the mean squared error of X for any P.