Which monotone transformations give a very loose confidence interval in transformed space?

Let $X$ be a random variable with $[l, u]$ as its 95% confidence interval. It is known that for any monotonically increasing function $f$, a 95% CI for random variable $ f(X)$ is $ [f(l), f(u)] $. However, there are examples here showing that this may not be the shortest CI. In other words, $f$ can give a loose CI in the transformed space.

So the question is which functions will give very loose CI, which is undesirable. I guess how tight the CI $ [f(l), f(u)] $ is also depends on the distribution of $X$. So let us assume that $X$ is normally distributed.

Also, is there a way to measure the tightness of the transformed CI, given the function $f$ and the distribution of $X$?

Cross Validated Asked on November 12, 2021

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