How to ascertain statistical significance for nonparametric tests if the sample size is small In literature about nonparametric rank tests in event studies, I often encounter a variety of test statistics and authors often attempt to convince their readers that their methods overcome certain issues that others have or that they have other desirable properties. Tests in the event study space usually take a (financial) abnormal return and aim to uncover whether that return is significantly different from zero, given an estimation period.
One thing that I have noticed is that very often, a test statistic is derived and it is either implicitly assumed or explicitly stated that said test statistic follows a certain distribution, asymptotically.
For example, the authors of this paper derive the following test statistic for their nonparametric test:
$t_{rank} = Z \sqrt{\frac{T-2}{T-1-Z^2}} \xrightarrow{d} t_{n-2}$ as $n \rightarrow \infty $
This is merely an example and the details of the statistic are not of uttermost importance.
My question is: Considering how prevalent these asymptotic properties for nonparametric tests for event studies are, I wondered how one is able to test the statistical significance of a test statistic of a nonparametric test when sample size $n$ is small.
 A: Following my Comment, here is an example of finding powers of two
tests by means of simulation:
Suppose you have two samples of size $n_1 = n_2 = 10$ from
normal distributions with $\mu_1 = 1, \mu_2=3$ and $\sigma_1=\sigma_2 =1.$ 
A two sample t test is the natural test to see if data are likely to show a significant difference between the means. What is the probability of 'detection'? This is the 'power' of the test. 
[Specifically, we want to know whether the test can use the two sample means to find a difference of 2 units (in either direction) between the two population means. We are willing to assume that the two groups have the same variance.] 
But would it be a bad mistake to use the nonparametric Wilcoxon signed rank test instead? What is the
power of this Wilcoxon to detect such a difference in locations? 
For significance level 5%, a simple simulation in R shows the pooled 2-sample t test has probability almost 0.99
of correctly rejecting $H_0$. (The exact value 0.9882 can be found
using the noncentral t distribution.)
set.seed(2020)
pv.t = replicate(10^5, 
     t.test(rnorm(10,1,1),rnorm(10,3,1),bvar.eq=T)$p.val)
mean(pv.t < .05)
[1] 0.98776      # aprx power for pooled t test

Similarly, simulation shows that the Wicoxon rank sum test (as implemented in R) has a slightly smaller
probability of rejection, about 0.98. So the penalty for choosing the Wilcoxon test is not great. (An exact power for this nonparametric test would be more difficult
to find analytically.)
pv.w = replicate(10^5, 
     wilcox.test(rnorm(10,1,1),rnorm(10,3,1),bvar.eq=T)$p.val)
mean(pv.w < .05) 
[1] 0.98108      # aprx power for Wilcoxon SR test

