Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions

TL;DR

This paper demonstrates that the geometric mean is an effective and unbiased statistic for estimating the scale of heavy-tailed coupled Gaussian distributions, especially in the presence of outliers.

Contribution

It introduces a new scale estimator based on the geometric mean, valid for coupled Gaussian distributions, improving robustness over traditional methods.

Findings

The scale equals the product of the generalized mean and the square root of the coupling.

The geometric mean provides an unbiased estimate with diminishing variance for large samples.

The estimator is effective across a broad range of coupling values.

Abstract

The geometric mean is shown to be an appropriate statistic for the scale of a heavy-tailed coupled Gaussian distribution or equivalently the Student's t distribution. The coupled Gaussian is a member of a family of distributions parameterized by the nonlinear statistical coupling which is the reciprocal of the degree of freedom and is proportional to fluctuations in the inverse scale of the Gaussian. Existing estimators of the scale of the coupled Gaussian have relied on estimates of the full distribution, and they suffer from problems related to outliers in heavy-tailed distributions. In this paper, the scale of a coupled Gaussian is proven to be equal to the product of the generalized mean and the square root of the coupling. From our numerical computations of the scales of coupled Gaussians using the generalized mean of random samples, it is indicated that only samples from a Cauchy distribution (with coupling parameter one) form an unbiased estimate with diminishing variance for large samples. Nevertheless, we also prove that the scale is a function of the geometric mean, the coupling term and a harmonic number. Numerical experiments show that this estimator is unbiased with diminishing variance for large samples for a broad range of coupling values.