Confidence interval of Inverse Gamma distribution

Question

Let's say I have a set of value that is inverse gamma distributed, how do I compute the 95% confidence interval?

Is there a formula so that I can apply to find the range of interval?

Answer 1

The cumulative distribution function is $\dfrac{\Gamma(\alpha, \beta/x)}{\Gamma(\alpha)}$ where the numerator is an incomplete Gamma function so one approach might be to find the values of $x$ which make this $0.025$ and $0.975$.

You will probably want to use a computer for the calculations. For example you could use the pscl library in R to give something like

> library(pscl)
> c(qigamma(0.025, alpha=5,beta=2), qigamma(0.975, alpha=5,beta=2))
[1] 0.1952822 1.2319167

though as the distribution is asymmetric you might prefer a highest density region

> igammaHDR(alpha=5,beta=2,content=0.95)
[1] 0.1467018 1.0294612