(a) Let Y = X2 + Z, where Z is a zero-mean unit variance Gaussian rv. Show that no unbiased estimate of X exists from observation of Y. Hint. Consider any x > 0 and compare with -x.
(b) Let Y = X + Z, where Z is uniform over (-1, 1) and X is a parameter lying in (-1, 1). Show that xˆ(y) = y is an unbiased estimate of x. Find a biased estimate xˆ1(y) for which |xˆ1(y) - x| ≤ |xˆ(y) - x| for all x and y with strict inequality with positive probability for all x ∈ (-1, 1).