A characterization of the one parameter exponential family of distributions by monotonicity of likelihood ratios

Summary Any one parameter exponential family of distributions has monotone likelihood ratios. As the product probabilities of n identical distributions of an exponential family form again an exponential family, it has monotone likelihood ratios for arbitrary n. Furthermore, the members of an exponential family are mutually absolutely continuous. In Part 1, we show that these properties uniquely characterize the exponential family. The application of this result to the theory of testing hypotheses (Part 2) shows that if a family of mutually absolutely continuous distributions has uniformly most powerful tests for arbitrary levels of significance, and arbitrary sample sizes, then it is necessarily an exponential family.The research was done while this author was a Visiting Professor in the Department of Statistics at the University of Chicago. It was supported by Research Grants Nos. NSF-G10368 and NSF-G21058 from the Division of Mathematical, Physical and Engineering Sciences of the National Science Foundation.