Independent Approximates Provide a Maximum Likelihood Estimate for Heavy-Tailed Distributions

Abstract

Heavy-tailed distributions are infamously difficult to estimate because their moments tend to infinity as the shape of the tail decay increases. Nevertheless, this study shows that a modified group of moments can be used to determine a maximum likelihood estimate of heavy-tailed distributions. These modified moments are determined from powers of the original distribution. Within nonextensive statistical mechanics, this has been referred to as the escort distribution. Here we clarify that this is the distribution of Independent-Equals, the number of independent random variables sharing the same state. The nth-power distribution is guaranteed to have finite moments up to n-1. Samples from the nth-power distribution are drawn from n-tuple Independent Approximates, which are the set of independent samples grouped into n-tuples and sub-selected to be approximately equal to each other. We show that Independent Approximates are a maximum likelihood estimator for the generalized Pareto and the Student's t distributions, which are members of the family of coupled exponential distributions. We use the first (original), second, and third power distributions to estimate their zeroth (geometric mean), first, and second power-moments, respectively. In turn, these power-moments are used to estimate the scale and shape of the distributions. The least absolute deviation criterion is used to select the optimal set of Independent Approximates. Estimates using higher powers and moments are possible, though the number of n-tuples that are approximately equal may be limited.