REFINEMENTS OF THE ARITHMETIC-GEOMETRIC MEAN INEQUALITY USING THE THIRD CENTRAL MOMENT VIA NEWTON’S IDENTITIES AND SYMMETRIC MEANS

Abstract

The classical Arithmetic-Geometric (A-G) mean inequality provides a basic ordering for positive real numbers, however it provides no means of quantifying how far apart the means are based on distributional properties. Classical refined inequalities only make use of the second central moment (variance) to describe the dispersion and is thus “shape blind” as it doesn’t capture the skewness of the data. In this paper, we examine an improved third-order refinement using Newton’s Iden tities for relating the elementary symmetric functions and the power sums. If we use the third part of the Maclaurin inequality chain, S1 ≥ S1/2 2 ≥ S1/3 3 ≥ ··· ≥ G, we will have a closed form which includes the third central moment. It is shown that this “penalty for asymmetry” greatly enhances the approximation of the A-G gap, especially for right-skewed distributions where the arithmetic mean is pulled up much further than the geometric mean. Numerically it is seen that our higher-order models cut the error more than 80% compared with just variance-based models for asymmetry. The higher accuracy leads to large benefits for numerical stability at high-performance computation and tail risk estimation in finance.