(i) Let X take on the values 1 and + 1 with probability 1...

(i) Let X take on the values θ − 1 and θ + 1 with probability 1 2 each. The problem of estimating θ with loss function L(θ,

d) = min(|θ − d|, 1)

remains invariant under the transformation gX = X +

c, ¯gθ = θ +

c, g∗d = d + c.

Among invariant estimates, those taking on the values X − 1 and X + 1 with probabilities p and q (independent of X) uniformly minimize the risk. (ii) That the conclusion of