Question: 6. Fully informative statistics. A statistic T is fully informative if for every decision problem the decision procedures based only on T form an essentially

6. Fully informative statistics. A statistic T is fully informative if for every decision problem the decision procedures based only on T form an essentially complete class. If 9 is dominated and T is fully informative, then T is sufficient. [Consider any pair of distributions Po, PI E 9 with densities Po' PI' and let gi = p;/(Po + PI)' Suppose that T is fully informative, and let do be the subfie1d induced by T. Then do contains the subfield induced by (go, gl) since it contains every rejection region which is unique most powerful for testing Po against PI (or PI against Po) at some level

a. Therefore, T is sufficient for every pair of distributions (Po, PI)' and hence by Problem 10 of Chapter 2 it is sufficient for 9.]

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