# The hypothesis test for independence based on a test statistic

Assume that $$\pi$$ is the coupling of probability measures $$\mu$$ and $$\nu$$ on $$[0,1]$$. The hypothesis test for independence is that $$H_0: \pi=\mu\times \nu \, , H_a: \pi\neq \mu\times \nu$$ The test statistic is $$W(\hat{\pi}^N)$$.

My question is how to get the power: $$\text{power}=1−P(\text{type II error})=1−P(H_0 \text{ accept}|H_0 \text{ false})$$ Can I take two dependent random data sets and for this one means conditional probability on $$H_0$$ false?

Also, I am confused about the meaning of this Corollary. Does it mean the $$P(\text{reject } H_0|H_0 \text{ is true})$$, which is the Type-1 error? But in the simulation result, the author calculates the power and assume the significance level $$\alpha=0.1$$.

In the simulations, the author computes the power function of the test, which, in general, is the probability of a test $$T$$ rejecting the null as a function of the distribution. Thus, it is a little different from the power. I.e., if the parameter for the distribution is $$\theta$$, then the power function $$\beta_T(\theta)$$ is given as $$\beta_T(\theta) = p(T=1|\theta)$$. In particular, it is defined for all distributions, those in $$H_0$$ and those in $$H_1$$.
The corollary just says that there is a constant $$C(\nu)$$ such that $$W$$ would asymptotically make sense as a test statistic for independence.