Uniformly Most Powerful Test - The Karlin-Rubin Theorem

The Karlin-Rubin Theorem

The Karlin-Rubin theorem can be regarded as an extension of the Neyman-Pearson lemma for composite hypotheses. Consider a scalar measurement having a probability density function parameterized by a scalar parameter θ, and define the likelihood ratio . If is monotone non-decreasing, in, for any pair (meaning that the greater is, the more likely is), then the threshold test:

\phi(x) = \begin{cases} 1 & \text{if } x > x_0 \\ 0 & \text{if } x < x_0 \end{cases}
where is chosen such that

is the UMP test of size α for testing

Note that exactly the same test is also UMP for testing

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