# Unbiased test

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A statistical test of size (level) $\alpha$, $0 < \alpha < 1$, for testing a compound hypothesis $H_0$: $\theta\in\Theta_0\subset\Theta$ against the compound alternative $H_1$: $\theta\in\Theta_1=\Theta\setminus\Theta_0$, whose power function $\beta({\cdot})$ satisfies $$\beta(\theta) \le \alpha \ \ \text{if}\ \ \theta \in \Theta_0 \,,$$ $$\beta(\theta) \ge \alpha \ \ \text{if}\ \ \theta \in \Theta_1 \ .$$
In many problems in statistical hypotheses testing there are no uniformly most-powerful tests. But if one restricts the class of tests, then there may be uniformly most-powerful tests in that class. If in the problem of testing the hypothesis $H_0$ against the alternative $H_1$ there is a uniformly most-powerful test, then it is unbiased (cf. Unbiased test), since the power of such a test cannot be less than that of the so-called trivial test whose critical function $\Phi({\cdot})$ is constant and equal to the size $\alpha$, that is, $\Phi(X) = \alpha$, where $X$ is the random variable whose realization is used to test the hypothesis $H_0$ against the alternative $H_1$.
Example. The sign test is uniformly most-powerful unbiased in the problem of testing the hypothesis $H_0$ according to which the unknown true value of the parameter $p$ of the binomial distribution is equal to $\frac12$ against the alternative $H_1$: $p\ne\frac12$.