4.1 Unbiasedness For Hypothesis Testing

• $\Theta \subseteq \mathbb{R}^{p}$,
• \(\Theta_{H}, \Theta_{K}\),
  • \(\Theta_{H} \cup \Theta_{K} = \Theta\),

Definition. Unbiased

• $\phi: \mathcal{X} \rightarrow [0, 1]$
  • test
• $\alpha \in [0, 1]$

test is said to be $\alpha$-unbiased test if

\[\begin{eqnarray} \beta_{\phi}(\theta) \le \alpha, & & \theta \in \Theta_{H} \nonumber \\ \beta_{\phi}(\theta) \ge \alpha, & & \theta \in \Theta_{K} \label{chap04_04_01_unbiasesed_test} . \end{eqnarray}\]

The first equation is error of the first kind. The second equation is power of test. Note that level-$\alpha$ test satisfies the first equation by definition.

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Remarks

Risk function $R(\theta, \delta_{\phi})$ in statistical test is defined as

\[\begin{equation} R(\theta, \delta_{\phi}) = \begin{cases} \mathrm{E}_{\theta} \left[ \phi \right] & (\theta \in \Theta_{H}) \\ 1 - \mathrm{E}_{\theta} \left[ \phi \right] & (\theta \in \Theta_{K}) \end{cases} . \nonumber \end{equation}\]

If the test is a unbiased test, we can re-write the definiont of the unbiasedness in termso of risk function;

\[\begin{eqnarray} R(\theta, \delta_{\phi}) & \le & \alpha, \quad (\theta \in \Theta_{H}) \nonumber \\ R(\theta, \delta_{\phi}) & \le & 1 - \alpha, \quad (\theta \in \Theta_{K}) . \nonumber \end{eqnarray}\]

また、$\forall \phi$について、$\beta_{\phi}$が$\theta$について連続で、\(\Theta_{H}^{f} \cap \Theta_{K}^{f} \neq \emptyset\)(これは、\(\Theta_{K}, \Theta_{H}\)が\(\Theta\)の分割であれば、成り立つ)とすれば

\[\begin{equation} \beta_{\phi}(\theta) = \alpha, \ \forall \theta \in \Theta_{H}^{f} \cap \Theta_{K}^{f}, \ \label{chap04_04_02_similar_on_boundary} \end{equation}\]

ただし、\(\Theta_{H}^{f}, \Theta_{K}^{f}\)は\(\Theta_{H}, \Theta_{K}\)の境界である。 実際、\(\Theta_{H} = (\Theta_{K})^{c}\)より、\(\forall \theta \in \Theta_{H}^{f}\)について、

\[\{\theta_{H, i}\}_{i \in \mathbb{N}} \subset \Theta_{H}, \ \{\theta_{K, i}\}_{i \in \mathbb{N}} \subset \Theta_{K}, \ \theta_{H, i} \rightarrow \theta, \ \theta_{K, i} \rightarrow \theta,\]

がとれる。 よって、

\[\begin{eqnarray} \beta_{\phi}(\theta_{H, i}) \le \alpha & \rightarrow & \beta_{\phi}(\theta) \le \alpha \nonumber \\ \beta_{\phi}(\theta_{K, i}) \ge \alpha & \rightarrow & \beta_{\phi}(\theta) \ge \alpha \nonumber \end{eqnarray}\]

よりOK。

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Definitions. similar

• \(\Theta^{\prime} \subset \Theta\),
• $c \in [0, 1]$

以下を満たす時、検定$\phi$は$\Theta^{\prime}$上、similar（相似）という。

\[\forall \theta \in \Theta^{\prime}, \ E_{\theta}[\phi] = c\]

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不偏検定は境界上相似である。 一方、相似検定を考えた時に、不偏検定となるようにできるか？ 次の定理から、境界上相似なUMP testは、不偏検定となることが分かる。

Lemma 4.1.1

• $\beta_{\phi}$
  • $\theta \in \Theta$について、連続
• $\alpha \in [0, 1]$
• $\phi_{0}$
  • \(\eqref{chap04_04_02_similar_on_boundary}\)を満たすtestに対するUMP test
  • 水準$\alpha$のtest

このとき、$\phi_{0}$はUMP不偏検定である。

proof.

まず、$\phi_{0}$は水準$\alpha$の検定であるから、定義より、

\[\forall \theta \in \Theta_{H}, \ \beta_{\phi}(\theta) \le \alpha\]

$\phi \equiv \alpha$も\(\eqref{chap04_04_02_similar_on_boundary}\)を満たすから、$\phi_{0}$がUMP testより、

\[\forall \theta \in \Theta_{K}, \ \alpha = \beta_{\phi}(\theta) \le \beta_{\phi_{0}}(\theta)\]

$\Box$

このことから、相似検定のみを考えれば良いことになる。

4.2 One-Parameter Exponential Families

Definition. Exponential familiy

• $\Theta \subset \mathbb{R}^{m}$,
• \(\mathcal{P} := \{P_{\theta}\}_{\theta \in \Theta}\),
• \(\theta = (\theta_{1}, \ldots, \theta_{m}) \in \Theta\),
• $g: \mathcal{X} \rightarrow \mathbb{R}$
  • measurable function
• $T_{i}: \mathcal{X} \rightarrow \mathbb{R}$
  • measurable function
• $\psi:\Theta \rightarrow \mathbb{R}$
  • real-valued funtion
• $a_{i}:\Theta \rightarrow \mathbb{R}$
  • real-valued funtion

$\mathcal{P}$ is said to be exponential family if

\[\theta \in \Theta, \ \frac{ P_{\theta} }{ d \mu } = \exp \left( \sum_{i=1}^{m} a(\theta)_{i} T(x) - \phi(\theta) \right) g(x) .\]

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In this section, we consider the case that $m = 1$, $a_{i}$ is identity map, and$\phi(\theta) = -\log(C(\theta))$. We denote $h=g$ to be consistent with the book. With these notation, the above equation can be written as

\[\theta \in \Theta, \ \frac{ P_{\theta} }{ d \mu } = C(\theta) \exp \left( \theta T(x) \right) h(x) .\]

• case1
  • By Corollary 3.4.1, there exists UMP test
  • \(H: \theta \le \theta_{0}\),
  • \(K: \theta > \theta_{0}\),
• case2: \(\theta_{1} < \theta_{2}\),
  • By Theorem 3.7.1, there exists UMP test
  • $H$: \(\theta \le \theta_{1}\) or \(\theta \ge \theta_{2}\),
  • \(K: \theta_{1} < \theta < \theta_{2}\),
• case3: \(\theta_{1} < \theta_{2}\),
  • \(H: \theta_{1} \le \theta \le \theta_{2}\),
  • $K$: \(\theta < \theta_{1}\) or \(\theta > \theta_{2}\),

We will show that there exists unbiased UMP test in the case3. We define test $\phi$ as

\[\begin{equation} \phi(x) := \begin{cases} 1 & T(x) < C_{1} \text{ or } T(x) > C_{2} \\ \gamma_{i} & T(x) = C_{1} \text{ or } T(x) = C_{2} \\ 0 & C_{1} < T(x) < C_{2} \\ \end{cases} \label{chap04_04_03_test} . \end{equation}\]

$\gamma_{i}$と\(C_{i}\)は以下を満たすようにきめる。

\[\begin{equation} \mathrm{E}_{\theta_{1}} \left[ \phi(X) \right] = \mathrm{E}_{\theta_{2}} \left[ \phi(X) \right] = \alpha \label{chap04_04_04} \end{equation}\]