Let x be a transformation of y, whose distribution is unknown. We derive an expansion formulating the expectations of x in terms of the expectations of y. Apart from the intrinsic interest in such a fundamental relation, our results can be applied to calculating E(x) by the low-order moments of a transformation which can be chosen to give a good approximation for E(x). To do so, we generalize the approach of bounding the terms in expansions of characteristic functions, and use our result to derive an explicit and accurate bound for the remainder when a finite number of terms are taken. We illustrate one of the implications of our method by providing accurate naive bootstrap confidence intervals for the mean of a fat-tailed distribution with an infinite variance, in which case currently-available bootstrap methods are asymptotically invalid and unreliable in finite sample.

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