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  • Karim M. Abadir (a1) and Adriana Cornea-Madeira (a2)

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.

Corresponding author
*Address correspondence to Karim M. Abadir, American University in Cairo and Imperial College London; e-mail:
Adriana Cornea-Madeira, University of York Management School, York, UK; e-mail:
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We thank Essie Maasoumi, Natalia Markovich, Peter Phillips, Robert Taylor, Michael Wolf, and three anonymous referees for their comments. This article was invited at the 2nd conference of the International Society of Non Parametric Statistics (Cádiz), 11th International Vilnius Conference on Probability Theory and Mathematical Statistics, 23rd Midwest Econometrics Meeting (Bloomington, Indiana), 4th French Econometrics Conference (Rennes), Conference on Fat Tails (Rennes), Katholieke Universiteit Leuven, LSE, Queen Mary, University of Bologna, University of Cyprus, and University of Oxford. This research is supported by the ESRC grant RES062230790 and by the British Academy’s PDF/2009/370.

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Econometric Theory
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