Consider i.i.d. observations

x_(1),dots,x_(n)N(\\\\theta ,\\\\sigma ^(2))

with parameter space

(\\\\theta ,\\\\sigma ^(2))inR\\\\times

R^(+)

, where

R^(+)=(0,\\\\infty )

. Show

\\\\bar{x}

is the minimax estimator for the loss

((hat(\\\\theta )-\\\\theta )^(2))/(\\\\sigma ^(2))

. In other words, it minimizes

sup_(\\\\theta inR,\\\\sigma ^(2))>0E_(\\\\theta ,\\\\sigma ^(2))(((hat(\\\\theta ))-\\\\theta )^(2))/(\\\\sigma ^(2))

.

Consider i.i.d. observations X1,,XnN(,2) with parameter space (,2)R R+, where R+=(0,). Show X is the minimax estimator for the loss (^)2/2. In other words, it minimizes supR,2>0E,22(^)2