Let Q run through the collection of all laws for which EQ f 2 ∞. For the U -statistic Un , show that (a) Un is the unique unbiased estimator T of g(Q) which is symmetric in X1,..., Xn, n ≥ m. Hint: If V = T - Un, EQ V = 0. If Q{xi } = pi , i = 1,..., r, EQ V is a symmetric polynomial S( p1,..., pr ), homogeneous of degree n and 0 for pi ≥ 0 (the restriction J,i pi = 1 can be removed). Show by induction on r that S ≡ 0, then that V ≡ 0. (b) Un is the unbiased estimator of g(Q) with smallest variance. Hint: For an unbiased estimator T with symmetrization W, W = Un by part (a).