A new look at q-exponential distributions via excess statistics

Abstract : Q-exponential distributions play an important role in nonextensive statistics. They appear as the canonical distributions, i.e. the maximum generalized q-entropy distributions under mean constraint. Their relevance is also independently justified by their appearance in the theory of superstatistics introduced by Beck and Cohen. In this paper, we provide a third and independent rationale for these distributions. We indicate that q-exponentials are stable by a statistical normalization operation, and that Pickands’ extreme values theorem plays the role of a CLT-like theorem in this context. This suggests that q-exponentials can arise in many contexts if the system at hand or the measurement device introduces some threshold. Moreover we give an asymptotic connection between excess distributions and maximum q-entropy. We also highlight the role of Generalized Pareto Distributions in many applications and present several methods for the practical estimation of q-exponential parameters.
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Jean-François Bercher, Christophe Vignat. A new look at q-exponential distributions via excess statistics. Physica A, Elsevier, 2008, 387 (22), pp.5422-5432. ⟨10.1016/j.physa.2008.05.038⟩. ⟨hal-00621920⟩

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