16. (i) Let X take on the values 8 - 1 and 8 + 1 with probability t each. The problem of estimating 8 with loss function L(8,

d) = min(18 - dl,l) remains invariant under the transformation gX = X +

c, g8 = 8 +

c, g*d = d +

c. Among invariant estimates, those taking on the values X-I and X + 1 with probabilities p and q (independent of X) uniformly minimize the risk. (ii) That the conclusion of Problem 15 need not hold when G is infinite follows by comparing the best invariant estimates of (i) with the estimate 81(x) which is X + 1 when X < 0 and X-I when X O.