Show that the $p$th upper (right tail) quantile of the Gumbel Distribution is given by, $z_p$:
$$z_p = \mu - \beta \log\{-\log(1-p)\}$$
Find MLEs of the 10 year return level and the 100 year return level. Find an expression for var$(z_p)$ to find an approximate 95% Confidence Interval for the $10$ year return level and the $100$ year return level.
(The $10$ and $100$ year return level have a value for $p$ of $0.1$ and $0.01$ respectively, it's a dataset based on rainfall. )
So for this, I have set the distribution function (for a Gumbel Distribution), $F(y;\mu,\beta)$, equal to $1-p$:
$$F(y;\mu,\beta)=\exp[-\exp\{- \frac{y-\mu} \beta \}]$$ $$1-p=\exp[-\exp\{- \dfrac{y-\mu}{\beta} \}]$$
Solve for $y$ and you get the required function, $z_p$. This part is fine.
If you run a gumbelFitin R (for the dataset we've been given) you get estimates for $\hat{\mu} = 3.869$ and $\hat{\beta} = 0.195$, which means I caclulated the $MLE_{10} = 4.1302$ and $MLE_{100}=4.3292$. This part is also fine.
Finding the expression for var$(z_p)$ is what I find difficult. I have the formula, I think:
$$\operatorname{var}\{z_p\} \approx \left(\frac{\partial z_p}{\partial \mu}, \frac{\partial z_p}{\partial \beta}\right) \hat{J}^{-1} \left(\frac{\partial z_p}{\partial \mu}, \dfrac{\partial z_p}{\partial \beta}\right)^T$$
What I dont know about this is what to use for $\hat{J}^{-1}$. It's an inverse matrix, yes, but with what elements?
Then how would I use $z_p$ to find a confidence interval? Perhaps it is something like this:
$$z_p \pm \operatorname{SE}(z_p)$$ where the standard error is $\sqrt{\operatorname{var}(z_p)}$.
I appreciate any insight into this.