Lecture 13. C) Karlin-Rubin Theorem

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Karlin-Rubin Theorem

In some cases, it is straightforward to derive the UMP test.

The Karlin-Rubin theorem states that the UMP level [math]\alpha[/math] test exists for one-sided testing,

[math]H_{0}:\theta\leq\theta_{0}[/math] vs. [math]H_{1}:\theta\gt \theta_{0}[/math]

if [math]f\left(\left.X\right|\theta\right)[/math] belongs to the exponential family and [math]\omega\left(\theta\right)[/math] is monotone (i.e., the pdf/pmf satisfies the monotone likelihood ratio property).

In this case, we reject [math]H_{0}[/math] when [math]\sum_{i=1}^{n}t\left(X_{i}\right)[/math] is large.