Is there a parametric trend test with provable power?

I have $$N$$ samples $$x_1,\dots,x_N$$ and I want to test following hypotheses.

The null hypothesis is that $$x_1,\dots,x_N$$ i.i.d. sample from a normal poplulation $$N(\mu_1,\sigma^2)$$, with unknown mean $$\mu_1$$ and known variance $$\sigma^2$$. The alternative hypothesis is that the joint distribution of $$x_1,\dots,x_N$$ is an $$N$$-dimensional normal distribution $$N(\vec{\mu}_2,\Sigma)$$. The mean vector has the form $$(a,\dots,a) + (\beta(1),\beta(2),\dots,\beta(N))$$ with unknown $$a$$ and know $$\beta(x)$$. The covariance matrix $$\Sigma$$ is known. Further more, the function $$\beta$$ is an increasing function on $$x$$.

I want to test these hypotheses with provable power, so I need to take a test and give a lower bound on the power of the test rather than just estimates the power of the test by Monte-Carlo simulation. But I don't know how to do it. Would you please give me some reference on the subject?

• You say $\Sigma$ is known. For a power calculation you need $\vec{\mu}_2$ too, which involves knowing $a$ and even then it may not be tractable Commented Nov 15, 2021 at 14:47
• @Henry Thanks for your kindly comment. I guess maybe there is a test with a lower bound on the power determined by the $\beta$ term, since the $\beta$ term is the only source of the trend. For example, by take $x_2-x_1,x_3-x_2,\dots,x_n - x_{n-1}$ I can ignore $a$. Commented Nov 15, 2021 at 14:53