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In the aforementioned book there is a statement:

(i) Supremum of the actual variance is infinite for any estimator whose value is always contained in convex hull of the observations.

If we take a mean, for an example, it's true that the mean will never be on the endpoints of the possible random variable value interval, so the supremum of the possible means will be infinity. But why the supremum of actual variance of a population mean will be infinity?

(ii) If the estimator is asymptotically normal, then the important central part of its distribution and confidence intervals for moderate confidence intervals can better be approximated in terms of the asymptotic variance than in terms of the actual variance.

So, what are the confidence intervals for the moderate confidence intervals? Why they can be better approximated with asymptotic variance?

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