It seems like a lot of examples for estimating the confidence intervals of a Gamma distribution, the parameter estimation involve one variable being known. I was wondering how to find a confidence interval for the MLE of the rate and scale parameters of a Gamma distribution.
Suppose we have $X_i\sim Gamma(\alpha, \beta)$, one can easily find the MLE estimators by solving: \begin{align*} \frac{\alpha}{\lambda} &= \bar{x} \\ \log{\alpha} - log{\bar{x}}- \psi(\alpha)+\overline{\log{x}}&=0 \end{align*} where $\psi(\alpha)=\frac{\partial}{\partial\alpha}\log{\Gamma(\alpha)}$ and we solve the second equation numerically to ultimately get estimators for $\alpha$ and $\beta$. My question is how exactly do we then get a confidence interval of $\alpha$ and $\beta$.
I am thinking that I can find the Fisher Information Matrix to get that $(\hat{\alpha}, \hat{\beta})\sim N([\hat{\alpha}, \hat{\beta}]^T, \mathbf{I^{-1}(\theta)})$, where $\mathbf{I^{-1}(\theta)}$ is the Fisher Information Matrix which can be found relatively easily. However, given that this Matrix has covariance terms, I was wondering how I would go about forming a confidence interval, or whether this is the correct approach to begin with.