For a multinomial distribution, let = i b i i , and suppose that

For a multinomial distribution, let γ = ∑_i b_i π_i, and suppose that π_i = f_i(θ) > 0, i = 1,..., I. For sample proportions {p_i}, let S = ∑_i b_i p_i. Let T = ∑_i b_i π̂_i, where π̂_i = f_i(θ̂), for the ML estimator θ̂ of θ.

a. Show that var(S) = [ ∑_i b_i² π_i – (∑_i b_i π_i­)²]/n.

b. Using the delta method, show var(T) ≈ [var(θ̂)][∑_i b_i f_i’(θ)]².

c. By computing the information for L(θ) = ∑_i n_i log[f_i(θ)], show that var(θ̂) is approximately [n∑_i (f_i’(θ))²/f_i(θ)]^–1.

d. Asymptotically, show that var[√n (T – γ)] ≤ var [√n (S – γ)]. [Show that var(T)/var(S) is a squared correlation between two random variables, where with probability π_i the first equals b_i and the second equals f_i’(θ)/f_i(θ).]