Let X1,..., Xn be i.i.d. P. Consider estimating (P) defined by (P) = E[h(X1,..., Xb)] , where

Let X1,..., Xn be i.i.d. P. Consider estimating θ(P) defined by

θ(P) = E[h(X1,..., Xb)] , where h is a symmetric kernel. Assume P is such that E[h2(X1,..., Xb)] < ∞, so that θ(P) is also well-defined. Let Un be the corresponding U-statistic defined by

(12.24). Let Pˆ

n be the empirical measure, and also consider the estimator

θ(Pˆ

n) = 1 nb n

i1=1

···n ib=1 h(Xi1 ,..., Xib ) .

Do √n[Un − θ(P)] and √n[θ(Pˆ

n) − θ(P)] converge to the same limiting distribution? If further conditions are needed, state them. Find the limiting behavior of n[Un − θ(Pˆ

n)]. Again, state any conditions you might need.