LINK OF MOMENTS BEFORE AND AFTER TRANSFORMATIONS, WITH AN APPLICATION TO RESAMPLING FROM FAT-TAILED DISTRIBUTIONS

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.