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The definition in the book I am reading says the following:

A test with power function $\beta(\theta)$ is unbiased if $\beta(\theta') \geq \beta(\theta'')$, $\forall \theta' \in \Theta_{0}^{\complement}, \theta'' \in \Theta_{0}$.

Now, $\beta(\theta)=\mathbb{P}_{\theta}(\mathbf{X}\in R)$ where $R$ is the rejection/critical region.We also know that:

$\mathbb{P}_{\theta}(\mathbf{X}\in R|\theta \in \Theta_{0})=\mathbb{P}(\text{Type I Error})$, and
$\mathbb{P}_{\theta}(\mathbf{X}\in R|\theta \in \Theta_{0}^{\complement})=1-\mathbb{P}(\text{Type II Error})$

That means we can also write:

$\beta({\theta'|\theta' \in \Theta_{0}})=\mathbb{P}(\text{Type I Error})$, and
$\beta({\theta''|\theta'' \in \Theta_{0}^{\complement}})=1-\mathbb{P}(\text{Type II Error})$

And from these, would it be right to say that a test is unbiased if

$$ \mathbb{P}(\text{Type I Error}) + \mathbb{P}(\text{Type II Error}) \leq 1 $$

If yes, how do I interpret that?

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2 Answers 2

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To ensure that $\beta(\theta_1) \geq \beta(\theta_0)$ for all $\theta_1 \in \Theta_{0}^{\complement}, \theta_0 \in \Theta_{0}$, it suffices to check that the inequality holds when the LHS is minimised and the RHS is maximised.

Notice that $\max_{\theta \in \Theta_{0}} \beta(\theta)=\max_{\theta \in \Theta_{0}}\mathbb{P}_\theta(\text{Type I error})=\alpha$, the size of the test.

You can, thus, rewrite the condition as $\beta(\theta_1) \geq \alpha$ for all $\theta_1 \in \Theta_{0}^{\complement}$. In words, when the alternative hypothesis is true, the power of the test is always greater than or equal to the size.

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  • $\begingroup$ +1. This can't be clearer (in OP's notations). $\endgroup$ Mar 27 at 11:37
  • $\begingroup$ @User1865345 Thanks! $\endgroup$ Mar 27 at 11:43
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Let me re-write the definition in a more familiar expression:

A size $\alpha$ test $\varphi$ is unbiased if

\begin{align}\mathbf E_\theta\varphi(X)&\leq \alpha,~~\theta\in\Theta_0,\\\mathbf E_\theta\varphi(X)&\geq \alpha,~~\theta\in\Theta_1.\end{align}

Here $\beta_\varphi(\theta) :=\mathbf E_\theta\varphi(X) $ is the power function of the test $\varphi.$

The interpretation is clear: the probability of $\varphi$ rejecting $\rm H_0$ when false is never smaller than that of rejecting of $\rm H_0$ when it is true.


Reference:

$\rm[I]$ Mathematical Statistics: A Decision Theoretic Approach, Thomas S. Ferguson, Academic Press, $1967, $ p. $224.$

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