Neyman-Pearson Theorem

Definition

The Neyman-Pearson theorem states that the Most Powerful Test for choosing between two simple hypotheses is based on the likelihood ratio.

Let $X_1,X_2,\dots ,X_n$ be random samples with PDF $f(x|\theta )$, where $\theta \in \Omega =\{{\theta }_0,{\theta }_1\}$, then the Likelihood Function of $X_1,X_2,\dots ,X_n$ is defined as $L(\theta |\mathbf{x})=\prod _{i=1}^{n}f(x_i|\theta )$ Consider the simple null hypothesis $H_0:\theta ={\theta }_0$ against the simple alternative hypothesis $\theta ={\theta }_1$ and define the likelihood ratio test statistic. $\Lambda (\mathbf{x})=\frac{L({\theta }_1|\mathbf{x})}{L({\theta }_0|\mathbf{x})}$

Let $k\in {\mathbb{R}}^{+}$, and $C$ be a subset of the sample space such that

• $\forall \mathbf{x}\in C^{\complement },\Lambda (\mathbf{x})\le k$
• $\forall \mathbf{x}\in C,\Lambda (\mathbf{x})\ge k$
• $P_{{\theta }_0}(\mathbf{X}\in C)=\alpha$ where ${\mathbf{x}}^{⊺}=(x_1,x_2,\dots ,x_n)$

Then, $C$ is a Best Critical Region of size $\alpha$ for the hypothesis test.