The mean squared error of an estimator is

. If is unbiased, then but in general MSE( . Consider the esti- ˆ

u) 5 V( ˆ

u) 1 (bias)2 MSE(ˆ

u) 5 V(ˆ

u), ˆ MSE( u ˆ

u) 5 E(ˆ

u 2 u)2

ˆ

u Use the fact that Y y iff each Xi y to derive the cdf of Y. Then show that the pdf of Y max(Xi) is

b. Use the result of part

(a) to show that the mle is biased but that (n  1)max(Xi

)/n is unbiased.

fY (y) 5 u nyn21 un 0 # y # u 0 otherwise mator , where S sˆ 2 sample variance. What value of