Penultimate gamma approximation in the CLT for skewed distributions

Department of Statistics and Insurance Science, University of Piraeus, Greece.
mbouts@unipi.gr

Received: 30 July 2014
Revised: 25 February 2015

Abstract

Sharp upper bounds are offered for the total variation distance between the distribution of a sum of n independent random variables following a skewed distribution with an absolutely continuous part, and an appropriate shifted gamma distribution. These bounds vanish at a rate O(n^-1) as n → ∞ while the corresponding distance to the normal distribution vanishes at a rate O(n^{− 1 / 2}), implying that, for skewed summands, pre-asymptotic (penultimate) gamma approximation is much more accurate than the usual normal approximation. Two illustrative examples concerning lognormal and Pareto summands are presented along with numerical comparisons confirming the aforementioned ascertainment.