8.1 Most Powerful Tests

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Recall 4.5 Introduction to Hypothesis Testing

Definition 8.1.1: Best critical region

Definition

Let $C$ : Subset of sample space

If

1. $P_{\theta’}[\mathbf{X}\in C]=\alpha$$
2. $ \forall A\subset C, P_{\theta’}[\mathbf{X}\in A]=\alpha\implies P_{\theta”}[\mathbf{X}\in C]\geq P_{\theta”}[\mathbf{X}\in A] $$

Then

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

Theorem 8.1.1: 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 $\Omega =\{\theta :\theta ={\theta }^{\prime },{\theta }^{\prime \prime }\}$
• $C$ : Subset of the sample space

If for any $k>0$

1. $ \frac{L(\theta’;\mathbf{x})}{L(\theta”;\mathbf{x})} \leq k, \quad \forall \mathbf{x}\in C $$
2. $ \frac{L(\theta’;\mathbf{x})}{L(\theta”;\mathbf{x})} \geq k, \quad \forall \mathbf{x}\in C^c $$
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 $H_1:\theta ={\theta }^{\prime \prime }$

Definition 8.1.2: Unbiased test

Definition

Let

• $X$ : Random variable with pdf/pmf $f(x;\theta ),\theta \in \Omega$
• ${\mathbf{X}}^{\prime }=(X_1,\dots ,X_n)$ : Random sample on $X$

Consider

• Hypotheses $H_0:\theta \in {\omega }_0$ versus $H_1:\theta \in {\omega }_1$
• A test with critical region $C$ and significance level $\alpha$

If $ P_{\theta}(\mathbf{X}\in C) \geq \alpha, \quad \forall \theta \in \omega_{1} $$

Then we say that this test is unbiased

Corollary 8.1.1

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 \leq \gamma_{C}(\theta”) $$
• The best test is an unbiased test