In an I J contingency table, let ij denote local odds ratio (2.10), and let

In an I × J contingency table, let θ_ij denote local odds ratio (2.10), and let θ̂_ij denote its sample value.

a. show that asymp. cov(√n log θ̂_ij, √n log θ̂_{i+1, j}) = –[π^–1_{i+1, j} + π^–1_{i+1, j+1}].

b. Show that asymp. cov(√n log θ̂_ij, √nlog θ̂_{i+1, j+1}) = π^–1_{i+1, j+1}.

c. When θ̂_ij and θ̂_hk use mutually exclusive sets of cells, show that asymp. cov(√n log θ̂_ij, √n log θ̂_hk) = 0.

d. State the asymptotic distribution of log θ̂_ij.