Having come across this discussion I'm raising the question on the back-transformed confidence intervals conventions.
According to this article the nominal coverage back-transformed CI for the mean of a log-normal random variable is:
$\ UCL(X)= \exp\left(Y+\frac{\text{var}(Y)}{2}+z\sqrt{\frac{\text{var}(Y)}{n}+\frac{\text{var}(Y)^2}{2(n-1)}}\right)$ $\ LCL(X)= \exp\left(Y+\frac{\text{var}(Y)}{2}-z\sqrt{\frac{\text{var}(Y)}{n}+\frac{\text{var}(Y)^2}{2(n-1)}}\right)$
/and not the naive $\exp((Y)+z\sqrt{\text{var}(Y)})$/
Now, what are such CIs for the following transformations:
- $\sqrt{x}$ and $x^{1/3}$
- $\text{arcsin}(\sqrt{x})$
- $\log(\frac{x}{1-x})$
- $1/x$
How about the tolerance interval for the random variable itself (I mean a single sample value randomly drawn from the population)? Is there the same issue with the back-transformed intervals, or will they have the nominal coverage?