Uniformly Most Powerful Test

Definition

A test defined by Uniformly Most Powerful Critical Region is called a uniformly most powerful (UMP) test

Simple vs Composite

Consider $H_0:\theta ={\theta }^{\prime }$ is a simple hypothesis and $H_1:\theta >{\theta }^{\prime },(\theta \ne {\theta }^{\prime },\theta <{\theta }^{\prime })$ is a composite hypothesis. Fix some ${\theta }^{\prime \prime }$ satisfying $H_1$ and obtain a most powerful test using Neyman-Pearson Theorem and generalize it to all ${\theta }^{\prime \prime }$ satisfying $H_1$ conditions

Composite vs Composite

Karlin-Rubin Theorem

Definition

Consider the hypothesis $H_0:\theta \le {\theta }^{\prime },(\theta \ge {\theta }^{\prime })$ and $H_1:\theta >{\theta }^{\prime }(\theta <{\theta }^{\prime })$, and assume that the likelihood ratio has MLR property in some Sufficient Statistic $Y=u(\mathbf{x})$ If the likelihood ratio is decreasing (increasing) in the statistic $Y=u(\mathbf{x})$ Then the Uniformly Most Powerful Test of level $\alpha$ for the hypothesis is rejecting $H_0$ if $Y\ge c,(Y\le c)$, where $c$ is determined by $\alpha =P_{{\theta }^{\prime }}(Y\ge c)$

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