Important Case: The Exponential Family
Although the Karlin-Rubin theorem may seem weak because of its restriction to scalar parameter and scalar measurement, it turns out that there exist a host of problems for which the theorem holds. In particular, the one-dimensional exponential family of probability density functions or probability mass functions with
has a monotone non-decreasing likelihood ratio in the sufficient statistic T(x), provided that is non-decreasing.
Read more about this topic: Uniformly Most Powerful Test
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