I have an MLE estimator which is asymptotically normally distributed with mean $\beta$ and variance $\beta^2/n$. How do I get an approximate confidence interval for this estimator?
I know usually two ways to do it: if we know the variance, we construct a confidence interval using normal quantiles, and if we don't know it, we plug in a sample standard deviation and use t-quantiles. But now the variance is not known but we also don't have a sample standard deviation at hand, so what to do?
What I could do is plug in the MLE estimator in my variance, and then I can construct a normal confidence interval using the standard $z$-quantiles, but how is this assumption warranted?