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Consider X2W in (10.8).
a. Use the linearization method of Section 9.1 to approximate V (ˆθ) in terms of V (Ṕij) and Cov(Ṕij , Ṕkl). Show that if we let yijk = 1 if observation k is in cell (i, j) and 0 otherwise, then Ṽ (ˆθ) = Ṽ (q-), where qk = Ṕ22y11k + Ṕ11y22k − Ṕ12y21k − Ṕ21y12k.
b. What is the Wald statistic, using the linearization estimate of V (ˆθ) in (a), when multinomial sampling is used? (Under multinomial sampling, V (Ṕij) = pij (1 − pij) /n and Cov(Ṕij , Ṕkl)=−pijpkl /n.) Is this the same as Pearson’s X2 statistic? |
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