Consider X 2 W in (10.8). 1. Use the linearization method of Section 9.1 to approximate V () in terms of V ( ij )

Consider X²_W in (10.8).

1. Use the linearization method of Section 9.1 to approximate V (ˆθ) in terms of V (Ṕ_ij) and Cov(Ṕ_ij , Ṕ_kl). Show that if we let y_ijk = 1 if observation k is in cell (i, j) and 0 otherwise, then Ṽ (ˆθ) = Ṽ (q^-), where q_k = Ṕ₂₂y₁₁k + Ṕ₁₁y₂₂k − Ṕ₁₂y₂₁k − Ṕ₂₁y₁₂k.

2. What is the Wald statistic, using the linearization estimate of V (ˆθ) in (a), when multinomial sampling is used? (Under multinomial sampling, V (Ṕ_ij) = p_ij (1 − p^ij) /n and Cov(Ṕ_ij , Ṕ_kl)=−p_ijp_kl /n.) Is this the same as Pearson’s X² statistic?