8.1 Most Powerful Tests

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Recall 4.5 Introduction to Hypothesis Testing

Definition 8.1.1: Best critical region

Let $C$ : Subset of sample space

If

1. $P_{{\theta }^{\prime }}[\mathbf{X}\in C]=\alpha$
2. $\forall A\subset C,P_{{\theta }^{\prime }}[\mathbf{X}\in A]=\alpha ⟹P_{{\theta }^{\prime \prime }}[\mathbf{X}\in C]\ge P_{{\theta }^{\prime \prime }}[\mathbf{X}\in A]$

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

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Theorem 8.1.1: Neyman-Pearson Theorem

Let

• $X_1,\dots ,X_n$ : Random sample with pdf/pmf $f(x;\theta )$
• $L(\theta ,\mathbf{x})$ : Likelihood function of $X_1,..,X_n$

Let

• ${\theta }^{\prime }$ and ${\theta }^{\prime \prime }$ be distinct fixed values of $\theta$ so that $\Omega =\{\theta :\theta ={\theta }^{\prime },{\theta }^{\prime \prime }\}$
• $k>0$

Let

• $C$ : 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. $\alpha =P_{H_0}[\mathbf{X}\in C]$

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 $H_1:\theta ={\theta }^{\prime \prime }$

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Definition 8.1.2: Unbiased test

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 $P_{\theta }(\mathbf{X}\in C)\ge \alpha ,\forall \theta \in {\omega }_1$

Then we say that this test is unbiased

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Corollary 8.1.1

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

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