Consider the null multinomial model, having the same probabilities {????j}

for every observation. Let ???? = ∑

j bj????j, and suppose that ????j = fj(????) > 0, j =

1,…,

c. For sample proportions {pj = nj∕N}, let S = ∑

j bjpj. Let T = ∑

j bj ̂????j, where ̂????j = fj(????̂), for the ML estimator ????̂ of ????.

a. Show that var(S) = [

∑

j b2 j ????j − (

∑

j bj

????j)

2]∕N.

b. Using the delta method, show var(T) ≈ [var(????̂)][∑

j bjf ′

j (????)]2.

c. By computing the information for L(????) = ∑

j nj log[fj

(????)], show that var(????̂)

is approximately [N ∑

j

(f ′

j (????))2∕fj(????)]−1.

d. Asymptotically, show that a consequence of model parsimony is that var[√

N(T − ????)] ≤ var[√

N(S − ????)].