# 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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