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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.

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  • $\begingroup$ Slutsky theorem $\endgroup$ – user45523 Dec 20 '20 at 8:51

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