Fisher Matrix and Parameter Confidence Interval

I am working with some data that fits a gamma distribution and would like to get confidence intervals for the parameters alpha and beta. I have found the general form for the fisher information matrix of a gamma here and have the inverse to get the correlation matrix, but was wonder about how to develop confidence intervals. I have two specific questions

1) Can I use the variance of the parameter to build confidence intervals like I would with a normal distribution, ie lower 95 CI = mean - 1.96*variance? I recall hearing that the information matrix tends towards normal as n => Inf.

2) If I wanted to know the confidence interval of the product of the two parameters, alpha * beta to get the mean, how could I do that? Can I do the same method as the previous question but just the the Var(A*B) = Var(A) + Var(B) + 2Cov(A,B)? So lower 95 CI = A*B - 1.96*Var(A,B).


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To answer question 1) for typical frequentists analysis the intervals 'typically' rely on asymptotics so what you're suggesting is essentially correct (you would want the standard error instead of the variance)

For question 2 that would be the variance of the sum - not the variance of the product. You would need to use the the delta method to get at the variance of more complicated functions (which the product is). You could alternatively just search out what the variance of the product is.