10. Pairwise sufficiency. A statistic T is pairwise sufficientfor 9 if it is sufficient for every pair...
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10. Pairwise sufficiency. A statistic T is pairwise sufficientfor 9 if it is sufficient for every pair of distributions in 9 . (i) If 9 is countable and T is pairwise sufficient for 9, then T is sufficient for 9 . (ii) If 9 is a dominated family and T is pairwise sufficient for 9, then T is sufficient for 9 . [(i): Let 9 = {Po, PI" " }, and let do be the sufficientsubfield induced by T. Let A = LCi Pi (Ci > 0) be equivalent to 9. For each j = 1,2, .. . the probability measure Aj that is proportional to (coin) Po + c/j is equivalent to (~)' ~} Thus by pairwise sufficiency, the derivative fj = dPo/[(coln) dPo + cj dPj ) ] is do·measurable. Let S, = (x :fj(x) = O} and S = Ui_ISj ' Then S edo, Po(S) = 0, and on !I"- S the derivative dPoldL'J=I Cj~ equals (L'j_11I fj) - I which is do·measurable. It then follows from Problem 2 that dA n dL Cj~ dPo }=o n dL Cj~ } =o dPo = ----=-- dA is also do·measurable. (ii): Let A = LJ=ICjPU be equivalent to 9 . Then pairwise sufficiency of T implies for any 80 that}dPu/(dPuo+ dA) and hence dPu/dA is a measurable function of T.]
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