3.36 Assume independence, and let pij = nij/n and tij = Pi+P+j•

a. Show that pij and tij are unbiased for πij = πi+N+j.

b. Show that var(pij) = πί+ π+j(1 – πί + π+j)/n.

c. Using E(pi+P+j)² = E(p²+)E(p) and E(p+ E(p+) = var(pi+) + [E(pi+)]², show that var(ij) = {πί+ Π+j [πί+(1 – π+j) + π+j(1 – π₁+)]}/n

+ni+(1 – πi+)π+j(1 – π+j)/n².

d. As n → ∞, show that lim var(√n t₁₁) ≤ lim var(√n pij), with equality only if πij = 1 or 0. Hence, if the model holds or if it nearly holds, the model estimator is better than the sample proportion.