Alternative way to characterize a q-gaussian distribution by a robust heavy tail measurement

The q-Gaussians are a class of stable distributions which are present in many scientific fields, and that behave as heavy tailed distributions for an especific range of q values. The identification of these values, which are used in the description of systems, is sometimes a hard task. In this work the identification of a q-Gaussian distribution from empirical data was done by a measure of its tail weight using robust statistics. Numerical methods were used to generate artificial data, to find out the tail weight – medcouple, and also to adjust the curve between medcouple and the q value. We showed that the medcouple value remains unchanged when the calculation is applied to data which have long memory. A routine was made to calculate the q value and its standard deviation, when applied to empirical data. It is possible to identify a q-Gaussian by the proposed methods with higher precision than in the literature for the same data sample, or as precise as found in the literature. However, in this case, it is required a smaller sample of data. We hope that this method will be able to open new ways for identifying physical phenomena that belongs to nonextensive frameworks.

💡 Research Summary

The paper proposes a novel, robust‑statistics‑based method for identifying q‑Gaussian distributions from empirical data by exploiting the medcouple (MC) statistic as a measure of tail weight. q‑Gaussians, which reduce to the normal distribution when q = 1 and exhibit heavy tails for 1 < q < 3, appear in many fields (physics, geophysics, finance, etc.). Traditional parameter estimation techniques (maximum likelihood, least‑squares fitting) often require very large samples and are highly sensitive to outliers or finite‑size effects, making the accurate determination of the non‑extensivity index q difficult.

The authors adopt the medcouple, originally introduced as a robust skewness estimator, because it is invariant under scale and location transformations and remains stable in the presence of up to about 12 % contamination. For symmetric distributions the left and right medcouple values coincide, so a single right‑medcouple (RMC) suffices to quantify tail heaviness.

Using the Mersenne‑Twister RNG in R, synthetic q‑Gaussian data are generated for a wide range of q values (mapped to the auxiliary variable Z = (q − 1)/(3 − q), spanning −1 < Z < 6). For each q, K = 2⁸ independent time series are created with lengths N = 2¹³, 2¹⁴, 2¹⁵. The robustbase package computes RMC for each series in O(N log N) time; the mean of the K RMC values, denoted m(q), serves as the empirical tail‑weight indicator.

A clear monotonic relationship between m and Z is observed. The authors fit this relationship with a hyperbolic‑tangent function of the form
 m(Z) = tanh