Link of Moments Before and After Transformations, with an Application to Resampling from Fat-Tailed Distributions
29 Pages Posted: 15 Oct 2011 Last revised: 28 Feb 2018
Date Written: 2012
We derive expansions of E(x) in terms of the moments of a transformation of x, in a more general context than Taylor expansions. Apart from the intrinsic interest in such a fundamental relation that links the moments of a variate and its nonlinear transformations, our results can be used in practice to approximate E(x) by the low-order moments of a transformation which can be chosen to give a good approximation for E(x). We generalize the approach of bounding the terms in expansions of characteristic functions, and use it to derive an explicit and accurate bound for the remainder in the moment expansion. We illustrate one of the implications of our method by providing accurate bootstrap condence intervals for the mean of a fat-tailed distribution with an infinite variance, in which case currently-available bootstrap methods are either asymptotically invalid or unreliable in finite sample.
Keywords: characteristic function, bootstrap, moments, confidence interval, infinite variance, heavy tails, stable law, domain of attraction, expansion of functions, remainder's bound, complex analysis
JEL Classification: C15, C12
Suggested Citation: Suggested Citation