# Intuitive understanding of size and power of a test

Let $$\mathcal{H}_0:\theta\in\Theta_0$$ and $$\mathcal{H}_1:\theta\in\Theta_1$$ be the null and alternative hypotheses. The power of a test $$\delta$$ is defined as: $$\pi_{\delta}:\Theta\to[0,1] \ s.t.\ \pi_{\delta}(\theta)=\mathbb{P}_{\theta}(\delta(X_1,...,X_n)=1)$$where $$\delta(X_1,...,X_n)=1$$ if it rejects $$\mathcal{H}_0$$ and 0 if it doesn't. Also, the size of the test $$\delta$$ is defined as: $$\alpha_{\delta}^*=\sup_{\theta\in\Theta_0}\pi_{\delta}(\theta)$$ I kind of understand these two concepts mathematically, but they're a bit dry to me yet. Does someone has an intuitive explanation for these concepts, or nice examples, interesting caveats, non-trivial but important remarks, etc. I really would like to have these concepts understood in a deeper way. Thanks in advance.

I think this is a case when the notation makes it much denser than it really is. $$\pi_\delta(\theta)$$ gives the probability of rejecting $$\mathcal H_0$$ given that the "truth" is $$\theta$$, and the size of the test is like the largest probability of rejecting given that $$\mathcal H_0$$ is true (hence the supremum only being over $$\theta\in\Theta_0$$). Also I think what you've written is the type I error rate not the power, since the power would be the probability of correctly rejecting when $$\theta\in\Theta_1$$.
We're only talking about varying $$\theta$$ here so the event $$\{\delta(\vec X) = 1\}$$ is fixed, so we could consider $$F := \{x \in \mathbb R^n : \delta(x) = 1\}$$ (or we could pull all the way back to the original sample space $$\Omega$$) and then $$\pi_\delta$$ gives us the size of this event for different $$\theta$$. And the size of the test then is the biggest measure we give to this event when we only consider $$\theta\in\Theta_0$$ (so the term "size" makes sense too).
I think this is a bit reminiscent of the difference between loss and risk in that $$\pi_\delta(\theta)$$ is a useful thing to know for one particular $$\theta$$, but we need to make a decision at the level of $$\Theta_0$$ which can contain uncountably many $$\theta$$, so we want to reduce this to a univariate summary (to take advantage of the total ordering of $$\mathbb R$$). Often the two candidates for doing this are integrating or optimizing and in this case optimizing turns out to be useful and intuitive (and the collection $$\{\pi_\delta(\theta) : \theta\in\Theta_0\}\subset \mathbb R$$ so the least upper bound property of $$\mathbb R$$ means we don't have to worry about the $$\sup$$ being well-defined).
One consequence of using the $$\sup$$ is that we only care about the largest $$F$$ can be. It doesn't matter if $$F$$ has a small measure for "most" $$\theta\in\Theta_0$$ (i.e. for most $$\theta\in\Theta_0$$ we are unlikely to mistakenly reject); if there are any $$\theta\in\Theta_0$$ that lead to a large probability of rejection then the test has a large size (i.e. type I error rate). This is a worst-case thing.