# Unnormalized t-test holding level when approximating asymptotic distribution

Let $$X$$ be some random variable with second moment and denote $$\mu = E(X)$$ and $$\sigma^2 = \textrm{Var}(X)$$. Let $$X_1, \dots X_n$$ be iid copies of $$X$$ and denote the usual empirical estimate of $$\sigma$$ as $$\hat{\sigma}$$. In the usual statistical literature we could test the hypothesis $$\mu = 0$$ using the test statistic

$$T = \frac{\frac{1}{\sqrt{n}} \sum_{i=1}^n X_i}{\hat{\sigma}}$$

which we know to be asymptotically standard normal under $$H_0$$ and we could calculate $$p$$-values and so on.

Let us assume instead that we were to consider

$$T^* = \frac{1}{\sqrt{n}} \sum_{i=1}^n X_i$$

this is still asymptotically normal but with unknown variance $$\sigma^2$$. We still have our estimate $$\hat{\sigma}$$ at our disposal. Could we use this to calculate quantiles of the approximated asymptotic distribution? Can we prove that this modified test holds level asymptotically using the approximated distribution to calculate $$p$$-values?

Side note: This is a dummy example, I'm well aware that this is not a smart idea but I'm working on a case where I've created a test statistic that depends on a parameter that can be estimated consistently and would like to prove that the test is well-behaved. In my case I cannot remove the parameters in an obvious way like it is done above.