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

See all articles by Karim M. Abadir

Karim M. Abadir

Imperial College Business School

Adriana Cornea

University of Exeter

Date Written: 2012

Abstract

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 con dence 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, con fidence interval, infi nite variance, heavy tails, stable law, domain of attraction, expansion of functions, remainder's bound, complex analysis

JEL Classification: C15, C12

Suggested Citation

Abadir, Karim M. and Cornea, Adriana, Link of Moments Before and After Transformations, with an Application to Resampling from Fat-Tailed Distributions (2012). Available at SSRN: https://ssrn.com/abstract=1942481 or http://dx.doi.org/10.2139/ssrn.1942481

Karim M. Abadir (Contact Author)

Imperial College Business School ( email )

South Kensington Campus
Exhibition Road
London SW7 2AZ, SW7 2AZ
United Kingdom

HOME PAGE: http://www3.imperial.ac.uk/portal/page?_pageid=61,629646&_dad=portallive&_schema=PORTALLIVE

Adriana Cornea

University of Exeter ( email )

Streatham Campus
Rennes Drive
Exeter, EX4 4ST
United Kingdom

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