Neyman-Pearson Theorem

Theorem

Let

• $X_1,\dots ,X_n$ : Random sample with pdf/pmf $f(x;\theta )$
• $L(\theta ,\mathbf{x})$ : Likelihood function of $X_1,..,X_n$
• ${\theta }^{\prime }$ and ${\theta }^{\prime \prime }$ be distinct fixed values of $\theta$ so that $\Omega =\{\theta :\theta ={\theta }^{\prime },{\theta }^{\prime \prime }\}$

If for $k>0$

• $C$ : Subset of the sample space such that
  1. $\frac{L({\theta }^{\prime };\mathbf{x})}{L({\theta }^{\prime \prime };\mathbf{x})}\le k,\forall \mathbf{x}\in C$
  2. $\frac{L({\theta }^{\prime };\mathbf{x})}{L({\theta }^{\prime \prime };\mathbf{x})}\ge k,\forall \mathbf{x}\in C^c$
  3. $\alpha =P_{H_0}[\mathbf{X}\in C]$ (significance level)

Then $C$ is a best critical region of size $\alpha$ for testing the simple hypothesis $H_0:\theta ={\theta }^{\prime }$ against the alternative simple hypothesis $H_1:\theta ={\theta }^{\prime \prime }$

Corollary

As in Neyman-Pearson Theorem,

Let

• $C$ : Critical region of the best test of $H_0:\theta ={\theta }^{\prime }$ versus $H_1:\theta ={\theta }^{\prime \prime }$
• $\alpha$ : Significance level of the test
• ${\gamma }_C({\theta }^{\prime \prime })=P_{{\theta }^{\prime \prime }}[\mathbf{X}\in C]$ : Power function of the test

Then

• $\alpha \le {\gamma }_C({\theta }^{\prime \prime })$
• The best test is an unbiased test