8.2 Uniformly Most Powerful Test

<< 8.1 Most Powerful Tests |

Definition 8.2.1

Definition

Let $C$ : Critical region of size $\alpha$ for testing the simple hypothesis $H_0$ against an alternative composite hypothesis $H_1$

If $C$ is a best critical region of size $\alpha$ for testing $H_0$ against each simple hypothesis in $H_1$

Then we say $C$ is a uniformly most powerful (UMP) critical region of size $\alpha$

Definition

Let $C$ : UMP critical region of size $\alpha$ for testing the simple hypothesis $H_0$ against an alternative composite hypothesis $H_1$

A test defined by $C$ is called a uniformly most powerful (UMP) test, with significance level $\alpha$, for testing the simple hypothesis $H_0$ against the alternative composite hypothesis $H_1$.

Definition 8.2.2: Monotone likelihood ratio (mlr)

Definition

Let $L(\theta ,\mathbf{x})$ : Likelihood function

If \frac{L(\theta_{1},\mathbf{x})}{L(\theta_{2},\mathbf{x})},\quad \forall \theta_{1}<\theta_{2} $$ is a monotone function ofy=u(\mathbf{x})$

Then we say $L(\theta ,\mathbf{x})$ has monotone likelihood ratio (mlr) in the statistic $y=u(\mathbf{x})$