Let Y1, Y2, . . . , Yn denote a random sample of size n from a population with a uniform distribution on the interval (, + 1) with density f(y) = 1, < y < + 1,

0, elsewhere.

Consider the sample mean Y , the smallest-order statistic Y(1) = min(Y1, Y2, . . . , Yn), and the largest-order statistic Y(n) = max(Y1, Y2, . . . , Yn). To estimate , we construct the following three different estimators using the above statistics: 1 = Y 0.5, 2 = Y(1), 3 = Y(n) 1.

(1) Prove that 1 is an unbiased estimator.

(2) Derive the bias of 2 and 3.

(3)Derive MSE(1) and MSE(2).