# Correct interpretation of an Unbiased Test

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?

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.

• +1. This can't be clearer (in OP's notations). Mar 27 at 11:37
• @User1865345 Thanks! Mar 27 at 11:43

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. $$224.$$