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My understanding is that the size $\alpha$ of hypothesis tests is defined as

$$ \alpha = \sup_{\theta \in \Theta_0} \mathbb{P}_\theta (X \in R) $$

where $\Theta_0$ is the subset of the parameter space associated with the null hypothesis $H_0: \theta \in \Theta_0$ and R is the rejection region such that

$$ \forall X \in R, \text{ reject } H_0 $$

So, in words, the size of hypothesis tests is the supremum of the probability to make type I error, i.e. rejecting $H_0$ when it is true. Correct?

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Broadly yes, but $\Theta_0$ is not "the null hypothesis", but contains the set of parameters for which the null is true.

For example, when you test a composite hypothesis about the mean that $H_0:\mu\leq0$, then the null is true for any nonpositive $\mu$. The definition of the size now says that (a little heuristically in view of the proper definition of a $\sup$) when we compute the type-I error probabilities for all these nonpositive $\mu$, the largest of these probabilities, so the maximum type-I error probability, is $\alpha$.

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    $\begingroup$ I edited the question to exactify the definition of $\Theta_0$ to "$\Theta_0$ is the subset of the parameter space associated with the null hypothesis $H_0: \theta \in \Theta_0$". So what does the qualifier "broadly" mean here? $\endgroup$
    – user78219
    Commented Nov 27, 2015 at 7:30
  • $\begingroup$ Broadly referred to that you broadly got it right. $\endgroup$
    – Christoph Hanck
    Commented Nov 27, 2015 at 7:32

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