4.30 Let X1, . . . , Xn be i.i.d. such that E(X1) = 0, E(X2 1) = 1, and E(X4 1) < ∞.

Consider approximation to the mean and variance of Y = n n +



n i=1 X2 i

using the delta method of Exercise 4.29.

(i) Let g(x1, . . . , xn) = n/(n +



n i=1 x2 i ). What are the approximations to the mean and variance of Y = g(X1, . . . , Xn)?

(ii) If we let g(t1, . . . , tn) = n/(n+



n i=1 ti ), and Ti = X2 i , 1 ≤ i ≤ n, what are the approximations to the mean and variance of Y = g(T1, . . . , Tn)?

(iii) How does the sample size n affect the approximation to E(Y )? In other words, does the accuracy of the approximation improve as n increases?

[Hint: First use the dominated convergence theorem (Theorem 2.16) to show that E(Y ) converges to a limit as n→∞.]

(iv) Which approximation [(i) or (ii)] do you think is better? Any general comment(s) on the use of the delta method in moment approximations?