Definition of Power and relationship with Type II error I've seen two definitions of Power:


*

*$P(\text{Rej. } H_0|\theta \in \Theta_1)$, from Wiki.

*$P_{\theta}(\text{Rej. } H_0)$, from Casella and Berger 'Statistical Inference'.


Which one is true? If we use the $2$nd, could we still state that power$=1-P(\text{type II error})$?
Any help would be appreciated.
 A: As far as I know, the power function $\pi (\theta)$ is defined for all $\theta \in \Theta$. Where, the difference stems from the interpretation/meaning of the function over various subsets of $\Theta$. Commonly, $\Theta = \Theta_0 \cup \Theta_1$, so, 
$$
\pi (\theta) = P(\text{reject}\, H_0 | \theta \in \Theta_1) = 1-\beta_{\theta}
$$
and 
$$
\pi(\theta) = P(\text{reject}\, H_0 | \theta \in \Theta_0) = \alpha_{\theta} .
$$
It means that for $H_0: \theta = \theta_1$ vs. $H_1:\theta = \theta_1$. $\pi(\theta)$ receives only two values, but for one sided hypothesis like $H_0:\theta_0 \ge \theta_1$ vs. $H_1:\theta_0 < \theta_1$, and, for instance, $\Theta = \mathbb{R}_{+}$. $\pi(\theta)$ will be strictly increasing function over $\mathbb{R}_{+}$, where $\lim_{\theta\to 0} \pi(\theta) = 0$ and $\lim_{\theta\to \infty} \pi(\theta) = 1$. In this case for UMP test you will get for the border line value $\pi(\theta_0) = \alpha $, this fact stems from the construction of UMP tests. Intuitively, it means that the least power of your test is attained at the border value, where for distant values of $\theta_1$ the power of your test will tend to $1$. As such for very small values of $\theta_0$ the probability of Type~I error will be very small.    
