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The geometric mean is shown to be an appropriate statistic for the scale of aheavy-tailed coupled Gaussian distribution or equivalently the Student's tdistribution. The coupled Gaussian is a member of a family of distributionsparameterized by the nonlinear statistical coupling which is the reciprocal ofthe degree of freedom and is proportional to fluctuations in the inverse scaleof the Gaussian. Existing estimators of the scale of the coupled Gaussian haverelied on estimates of the full distribution, and they suffer from problemsrelated to outliers in heavy-tailed distributions. In this paper, the scale ofa coupled Gaussian is proven to be equal to the product of the generalized meanand the square root of the coupling. From our numerical computations of thescales of coupled Gaussians using the generalized mean of random samples, it isindicated that only samples from a Cauchy distribution (with coupling parameterone) 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 thisestimator is unbiased with diminishing variance for large samples for a broadrange of coupling values.

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