1.19 If the X’s are as in Theorem 1.1 and if the first five derivatives of h exist and the fifth derivative is bounded, show that E[h(X¯ )] = h(ξ ) +

1 2

h σ2 n +

1 24n2 [4hµ3 + 3h(iv)

σ4

] + O(n−5/2

)

and if the fifth derivative of h2 is also bounded var[h(X¯ )] = (h2

)

σ2 n +

1 n2 [h

hµ3 + (h

h +

1 2

h2

)σ4

] + O(n−5/2

)

where µ3 = E(X − ξ )

3.

[Hint: Use the facts that E(X¯ − ξ )

3 = µ3/n2 and E(X¯ − ξ )

4 = 3σ4/n2 + O(1/n3).]

1.20 Under the assumptions of the preceding problem, carry the calculation of the variance (1.16) to terms of order 1/n2, and compare the result with that of the preceding problem.