I have two probabilities $p$ and $q$. $p>q$, and they aren't correlated. I'm going to calculate $i$ such that $p^i=q$, which is easily done as $\log_p(q)$.
Now, I'd like to also calculate a confidence interval for $i$, which is necessarily going to be a function of both $p$ and $q$'s confidence intervals. My first approach was to do
p.min <- qbeta(0.025, 152, 29) p.max <- qbeta(0.975, 152, 29) q.min <- qbeta(0.025, 37, 19) q.max <- qbeta(0.975, 37, 19) ## q.max and p.min have the smallest difference i.min <- log(q.max, base = p.min) ## q.min and p.max have the largest difference i.max <- log(q.min, base = p.max)
But it occurs to me that 95% confidence intervals for $p$ and $q$ independently probably produces too large a confidence interval for $i$, because the joint confidence interval for $p$ and $q$ will be narrower.
So, how do I go about figuring out the joint confidence interval of $p$ and $q$. They're uncorrelated, which should make things easier. It is as simple as narrowing the quantiles in
qbeta()? By how much?