Confidence Intervals affected by type of distribution

How is the accuracy of confidence intervals affected by the population of interests distribution? I'm having some trouble wrapping my head around this.

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Is a confidence statement more accurate when dealing with standard normal distributions or uniform distributions? Does it make any difference?


TS Contributor
I think the point of interest should be put in the opposite - when you have a certain kind of consistent point estimator which achieve asymptotic normality, then you may construct an approximate confidence interval based on the normal pivot, which does not affected by the original underlying distribution.