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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 $\alpha=0.1$.

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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.

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