Delta method for non-normal variables

Question

Is the delta method valid also for non-normal variables?

Claim:

Let $\sqrt{n}(\hat{X}_n-\theta) \xrightarrow{d} \hat{f} $.

With $\hat f$ having a finite distribution.

Then for every $g$ such that $g'(\theta)$ is nonzero we have:

$\sqrt{n}(g(\hat{X}_n)-g(\theta)) \xrightarrow{d} g'(\theta)\hat{f} $.

It seems to me that the standard proof:

applies also here, the only point that we need to assess that $\hat{X}_n \xrightarrow{P} \theta$.

I have also the impression that also $\sqrt{n}$ can be substituted by any increasing function of $n$.

Am I thinking wrong? Where can I find such generalizations of the delta method?

score 3 · Answer 1 · answered May 13 '17 at 06:49

3

The generalization is true and the standard reasoning presented in the Wikipedia page can be applied.

Confirmation can be found for example in these notes:

answered May 13 '17 at 06:49

Thomas

1 Answers1