= Statistical test
{title2=$\varphi(x)\in[0,1]$}
A statistical test is a measurable rule for deciding whether to reject a <null hypothesis>. Allowing randomisation, it is represented by a function $\varphi(x)\in[0,1]$ giving the conditional rejection <probability> after observing $x$. A deterministic test takes only values zero and one. Under a simple null, $E_0\varphi$ is its <size of a statistical test>; under an alternative, $E_1\varphi$ is its <statistical power>. Randomisation on a <likelihood ratio> equality set allows a <Neyman-Pearson lemma> threshold test to attain a prescribed size even for discrete data.
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