Link of moments before and after transformations, with an application to resampling from fat-tailed distributions

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JournalEconometric Theory
DateAccepted/In press - 28 Apr 2018
DateE-pub ahead of print - 4 Jun 2018
DatePublished (current) - 1 Jun 2019
Issue number3
Volume35
Number of pages23
Pages (from-to)630-652
Early online date4/06/18
Original languageEnglish

Abstract

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 is taken. We illustrate one of the implications of our method by providing accurate naive bootstrap confidence intervals for the mean of any fat-tailed distribution with an infinite variance, in which case currently available bootstrap methods are asymptotically invalid or unreliable in finite samples.

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