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

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  • Nelson, Kenric P.
  • Kon, Mark A.
  • Umarov, Sabir R.

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.

Suggested Citation

  • Nelson, Kenric P. & Kon, Mark A. & Umarov, Sabir R., 2019. "Use of the geometric mean as a statistic for the scale of the coupled Gaussian distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 515(C), pages 248-257.
    • Handle: RePEc:eee:phsmap:v:515:y:2019:i:c:p:248-257
    DOI: 10.1016/j.physa.2018.09.049
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    References listed on IDEAS

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    1. Nelson, Kenric P. & Umarov, Sabir, 2010. "Nonlinear statistical coupling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(11), pages 2157-2163.
    2. Nelson, Kenric P. & Umarov, Sabir R. & Kon, Mark A., 2017. "On the average uncertainty for systems with nonlinear coupling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 30-43.
    3. Hasegawa, Yoshihiko & Arita, Masanori, 2009. "Properties of the maximum q-likelihood estimator for independent random variables," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(17), pages 3399-3412.
    4. Vignat, C. & Plastino, A., 2009. "Why is the detection of q-Gaussian behavior such a common occurrence?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(5), pages 601-608.
    5. Nelson, Kenric P., 2015. "A definition of the coupled-product for multivariate coupled-exponentials," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 422(C), pages 187-192.
    6. Oikonomou, Th., 2007. "Tsallis, Rényi and nonextensive Gaussian entropy derived from the respective multinomial coefficients," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 386(1), pages 119-134.
    7. Borges, Ernesto P., 2004. "A possible deformed algebra and calculus inspired in nonextensive thermostatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 95-101.
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    Cited by:

    1. Nelson, Kenric P., 2022. "Independent Approximates enable closed-form estimation of heavy-tailed distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).
    2. Zubillaga, Bernardo J. & Vilela, André L.M. & Wang, Chao & Nelson, Kenric P. & Stanley, H. Eugene, 2022. "A three-state opinion formation model for financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 588(C).

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