Test Quality Function

Definition

Let $\varphi$ be a test, and $p_{\theta }$ be a probabilistic model parametrized by $\theta$, and $E_{p_{\theta }}$ the expectation of a random variable under $p_{\theta }$. The function defined as

q:&\Theta&\to&[0,1]\\ &\theta&\mapsto&E_{p_\theta(y)}(\phi(y)) \end{array}$$ is said to be a _test quality function_. > [!note] Remark > > The test quality function maps the parameter $\theta$ in the value given by the [[Expected Value|expectation]] of the test $\phi$ under the probabilistic model $p_\theta(y)$. > > > The definition of the test quality function is motivated by the value it assumes for $\theta\in\Theta_0$ and $\theta\in\Theta_1$: because the [[random variable]] $\phi$ only takes on values in $\{0, 1\}$, the expected value $E_{p_\theta(y)}(\phi(y))$ is identical to the [[Probability Distribution|probability]] of the event $\phi(y)=1$ under $p_\theta(y)$. Thus, for $\theta\in\Theta_0$, the test quality function returns the size of the test (Eq. [[Type-I Error Rate#^testsize|(test size)]]) and for $\theta\in\Theta_1$, the test quality function returns the power of the test (Eq. [[Type-II Error Rate#^testpower|(testpower)]]). ## References [[D. Ostwald - The general linear model 20_21|D. Ostwald - The general linear model 20_21 pp 118]]