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.


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