(i) Let X take on the values 1 and + 1 with probability 1 2 each. The problem of estimating with

(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, g¯θ = 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