Uniformly Most Powerful Test

A Uniformly Most Powerful Test is a statistical hypothesis testing that has the greatest statistical power over all possible tests of a given size.

• AKA: UMP, Best Critical Region.
• See: Neyman–Pearson Lemma, Statistical Hypothesis Test, Karlin–Rubin Theorem.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Uniformly_most_powerful_test Retrieved 2016-09-05
  • In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power 1 − β among all possible tests of a given size α. For example, according to the Neyman–Pearson lemma, the likelihood-ratio test is UMP for testing simple (point) hypotheses.

Let [math]\displaystyle{ X }[/math] denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability mass functions [math]\displaystyle{ f_{\theta}(x) }[/math], which depends on the unknown deterministic parameter [math]\displaystyle{ \theta \in \Theta }[/math]. The parameter space [math]\displaystyle{ \Theta }[/math] is partitioned into two disjoint sets [math]\displaystyle{ \Theta_0 }[/math] and [math]\displaystyle{ \Theta_1 }[/math]. Let [math]\displaystyle{ H_0 }[/math] denote the hypothesis that [math]\displaystyle{ \theta \in \Theta_0 }[/math], and let [math]\displaystyle{ H_1 }[/math] denote the hypothesis that [math]\displaystyle{ \theta \in \Theta_1 }[/math].

The binary test of hypotheses is performed using a test function [math]\displaystyle{ \phi(x) }[/math].

[math]\displaystyle{ \phi(x) = \begin{cases} 1 & \text{if } x \in R \\ 0 & \text{if } x \in A \end{cases} }[/math]

meaning that [math]\displaystyle{ H_1 }[/math] is in force if the measurement [math]\displaystyle{ X \in R }[/math] and that [math]\displaystyle{ H_0 }[/math] is in force if the measurement [math]\displaystyle{ X \in A }[/math]. Note that [math]\displaystyle{ A \cup R }[/math] is a disjoint covering of the measurement space.

(...) A test function [math]\displaystyle{ \phi(x) }[/math] is UMP of size [math]\displaystyle{ \alpha }[/math] if for any other test function [math]\displaystyle{ \phi'(x) }[/math] satisfying

[math]\displaystyle{ \begin{array}[b]{l} \sup_{\theta\in\Theta_0}\; \operatorname{E}_\theta\phi'(X)=\alpha'\leq\alpha=\sup_{\theta\in\Theta_0}\; \operatorname{E}_\theta\phi(X)\,\\ \text{we have}\; \forall \theta \in \Theta_1, \quad \operatorname{E}_\theta\phi'(X)=1-\beta'\leq 1-\beta=\operatorname{E}_\theta\phi(X)\end{array} }[/math]

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