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Suppose, that I'm conducting a simulation study that compares different confidence interval construction methods. If I do 1000 simulation runs, then the acceptable range for coverage level is $95\pm\sqrt{(95\times 5)/1000}=(93.6\%, 96.4\%)$.

But how can we compare the tail errors? Ideally, we want to get both tail errors are equal to $2.5\%$, but in simulation they are not going to be balanced. I would like to construct the confidence interval around difference of tail errors $t_u-t_l$, such that if the difference between tail errors falls within this interval then we can say that tail errors are similar and method constructs accurate confidence interval. Is it possible?

As I understand, the tail errors are dependent, so simple confidence interval for difference of two proportions is not a good idea.

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