Consider the model for a 2 2 table. 11 = 2 , 12

Consider the model for a 2 × 2 table. π₁₁ = θ², π₁₂ = π₂₁ = θ(1 – θ), π₂₂ = (1 – θ)², where θ is unknown (Problems 3.31 and 10.34).

a. Find the matrix A in (14.14) for this model.

b. Use A to obtain the asymptotic variance of θ̂. (As a check, it is simple to find it directly using the inverse of – E∂²L/∂θ², where L is the log likelihood.) For which θ value is the variance maximized? What is the distribution of θ̂ if θ = 0 or θ = 1?

c. Find the asymptotic covariance matrix of √n π̂.

d. Find df for testing fit using X².