I am working on this problem of income distribution.

I am assuming that my income data $X_i$ is Pareto : $f(x_i;\alpha) = {\alpha \over x_0}({x_i \over x_0})^{-(\alpha + 1)}$

I found my MLE estimator for $\alpha$ st: $\hat{\alpha} = ({1\over n}\sum_{i = 1}^n ln(X_i/x_0)^{-1}$ and with fisher information that its standard error is $\hat{se}(\hat{\alpha}) = \hat{\alpha}/\sqrt{n}$

Now the Pietra index with a Pareto assumption is as follows: $P(\alpha) = {(\alpha -1) ^ {\alpha-1} \over \alpha^\alpha}$

Now I estimated my Pietra index with the $\hat{\alpha}$ and am asked to find the standard error.
I am advised to use Delta Method.
I have my data $X_i$ which is pareto and constant value $\hat{\alpha}$ so I would have $a_i(X_i-\hat{\alpha}) \rightarrow^d somepareto $
However I am quite confused as to what $a_i$ would be, as we need to diverge to $infinity$ and what is the precise distribution this would converge to. And would the function to use the the pietra index?

Any hints or guidance are appreciated. (this is for problem sets in a statistics course)


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