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 P is dominated and T is fully informative, then T is sufficient.

[Consider any pair of distributions P0, P1 ∈ P with densities p0, p1, and let gi =

pi/(p0 + p1). Suppose that T is fully informative, and let A be the subfield induced by T . Then A contains the subfield induced by (g0, g1) since it contains every rejection which is unique most powerful for testing P0 against P1 (or P1 against P0)

at some level α. Therefore, T is sufficient for every pair of distributions (P0, P1), and hence by