UMP test and non-decreasing power function Let $\phi$ be a UMP test for $H_o: \theta \leq \theta_0$ and $H_1: \theta > \theta_0$. Let its power function, $E_\theta(\phi)$, be differentiable w.r.t. $\theta$.
Show the power function is non-decreasing in $\theta$.
I can understand why size should always be less than power in this case, but why does it have to be non-decreasing?

I am going to assume that $\mathcal F=\{f_{\theta}:\theta\in \Omega\}$ is a family of identifiable distributions having monotone likelihood ratio in some statistic $T(x)$. The Karlin-Rubin theorem then states that for testing $H_0:\theta\le \theta_0$ against $H_1:\theta>\theta_0$, there exists a UMP size $\alpha$ test of the form $\phi=\mathbf1_{\{T>c\}}+\gamma\mathbf1_{\{T=c\}}$ for every $\alpha\in (0,1)$.
Now consider testing $H:\theta=\theta_1'$ against $K:\theta=\theta_2'$ where $\theta_2'>\theta_1'$. Then $\phi$ is most powerful for this testing problem. So it is also an unbiased test (its power is at least its size), i.e. $E_{\theta_2'}\phi\ge E_{\theta_1'}\phi$. In fact, strict inequality holds due to identifiability of $\mathcal F$: $$E_{\theta_2'}\phi> E_{\theta_1'}\phi$$
As $\theta_1'$ and $\theta_2'$ are arbitrary, the above holds for every $\theta_2'>\theta_1'$. This makes the power function $E_{\theta}\phi$ strictly increasing in $\theta$. For details, you can refer to page 5 of the lecture notes here.
I don't think differentiability of power function is needed for proving this under the above setup.
