# Approximating standard error that contains a parameter, by replacing the parameter with its estimate

I am a bit confused about the following step I have seen in the stats literature which seems to me a bit circular.

Say you are approximating the standard error of the MoM estimate of an exponential distribution with parameter $$\lambda$$.

You can show using the Delta method that the MoM estimator (which is $$\frac{1}{\bar{X}}$$) will be approximately equal to $$\frac{\lambda}{\sqrt{n}}$$ (in fact you can show more: it is asymptotically normal).

At this point since the standard error depends on the unknown parameter $$\lambda$$ some resources recommend replacing it with its estimate $$\frac{1}{\bar{X}}$$, which gives that the approximate standard error of the MoM estimate is $$\frac{1}{\bar{X} \sqrt{n}}$$.

What is the formal justification for this last step? It does make sense to me on an intuitive level, but I am interested to work it out rigorously.

• Slutsky theorem – user45523 Dec 20 '20 at 8:51