Abstract
exist and are continuous in ~ for all 0 e P.. Barankin-Katz [2] t rea ted sufficient statistics of minimal dimension. Barankin-Maitra [3] considered "Fisher-Darmois-Koopman-Pi tman t h e o r e m " it asserts t ha t the existence of a sufficient statistic leads to an exponential family for the case f(x; O)=f,(z,; 0) . . . f,,(z,,; O), where f~(z; 0)'s are not necessarily identical. These results were obtained under a regular i ty condition on 0; bu t as Barankin [1] recognized, and as we shall show below, they are obtainable wi thout any assumption on D. Moreover, a global result is obtained for an extension of the above theorem wi thout analytici ty of f j (x; 0)'s.
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