Maximum Likelihood Estimation for q-Exponential (Tsallis) Distributions

Tsallis et al. proposed these distributions to handle statistical-mechanical systems with long-range interactions, necessitating (it is claimed) a non-extensive generalization of the ordinary Gibbs-Shannon entropy. Following Jaynes’s procedure of maximizing an entropy subject to constraints on expectation values [2], they got the q-exponential distributions, in which κ enforces the constraints, and q measures the departure from extensivity, Boltzmann-Gibbs statistics being recovered as q → 1. Tsallis’s ideas about non-extensive entropy and its possible applications, in and out of statistical mechanics, have attracted intense (not to say “extensive”) interest in physics; the bibliography at http://tsallis.cat.cbpf.br/biblio.htm has over 2000 entries. They are also quite controversial (see, e.g., Refs. [3, 4, 5, 6, 7, 8], the replies by Tsallis and others, and in some cases the replies to the replies). Whether or not the critics are correct, however, q-exponentials are still valid probability distributions, and can usefully describe some empirical phenomena. To this end, in a recent paper Douglas R. White et al. pose the problem of estimating the parameters q and κ from data by the method of maximum likelihood [9]. This note solves that problem. I first reparameterize Eq. 1 to simplify estimation and emphasize links to Pareto distributions. I then rehearse the math of finding the maximum likelihood estimator (MLE) for the q-exponential distribution, discussing its accuracy and precision, and adjustments for data in which samples below a fixed threshold are all dropped (“censoring”). I compare maximum-likelihood estimates to those found by the current practice of curve-fitting; the latter are inferior. Finally, I discuss testing the assumption that the data are q-exponentially distributed. Code implementing the MLE for q-exponentials is available at