# Power of a statistic respect to significance level

Let $$Z_{1},\ldots, Z_{n}$$ random variables i.i.d with symmetric continuous distribution in $$\theta$$.

We consider hypothesis test $$\begin{array}{rcl} H_{0}: && \theta=\theta_{0} \\ H_{a}: && \theta>\theta_{0} \\ \end{array}$$

Let $$T_{\theta}$$ be a statistic, and let $$\alpha=\mathsf{P}(T_{\theta_{0}}\geq c | H_{0} \mbox{ true})$$ is significance level, that is, reject the null hypothesis if $$T_{\theta_{0}}(Z)\geq c$$.

My question: What is the power of $$T$$?

Remark: We remember that

Definition. The power of a hypothesis test is the probability of making the correct decision if the alternative hypothesis is true. That is, the power of a hypothesis test is the probability of rejecting the null hypothesis $$H_{0}$$ when the alternative hypothesis $$H_{a}$$ is the hypothesis that is true.

I know that $$\textrm{power}=1-\mathsf{P}(T_{\theta_{a}}\geq k|H_{a}\mbox{ is true})=\mathsf{P}(T_{\theta_{a}}\leq k|H_{a}\mbox{ is true})$$ where $$k$$ depend of $$c$$. I do not understand as from $$c$$ I can get $$k$$ in general. For example, for statistic $$T_{\theta}=\frac{\sqrt{n}(\overline{Z}-\theta)}{s}$$ It is easy to see the dependency of $$c$$ (see this link). But it is not easy for the case of the signed statistic $$B=\sum_{i=1}^{n}\psi(Z_{i}-\theta) \quad \mbox{ where } \quad \psi(x):=\left\{\begin{array}{ll} 1 & x>0\\ 0 & x\leq 0 \end{array} \right.$$ or the Wilcoxon signed rank statistic $$W^{+}=\sum_{i=1}^{n}\psi(Z_{i}-\theta)R^{+}_{i}$$ where $$R^{+}=(R^{+}_{1},\ldots,R^{+}_{n})$$ is the rank vector of $$|Z_{1}|,\ldots,|Z_{n}|$$. I really have not been able to see that dependence in these cases.

In general, we have the relationship $$\alpha \to c = k \to \beta_{\alpha}$$, where $$\alpha$$ is the significance level of the test and $$\beta_{\alpha}$$ is the power of the test at the significance level $$\alpha$$. This allows us to consider the power as a function of the level of the test, which is useful when evaluating the actual usefulness of tests done at various significance levels. So, the dependence of $$k$$ on $$c$$ is one of identity: $$k = c$$.