Independent Approximates enable closed-form estimation of heavy-tailed distributions

A new statistical estimation method, Independent Approximates (IAs), is defined and proven to enable closed-form estimation of the parameters of heavy-tailed distributions. Given independent, identically distributed samples from a one-dimensional distribution, IAs are formed by partitioning samples into pairs, triplets, or nth-order groupings and retaining the median of those groupings that are approximately equal. The pdf of the IAs is proven to be the normalized n^th power of the original density. From this property, heavy-tailed distributions are proven to have well-defined means for their IA pairs, finite second moments for their IA triplets, and a finite, well-defined (n-1)^th moment for the nth grouping. Estimation of the location, scale, and shape (inverse of degree of freedom) of the generalized Pareto and Student's t distributions are possible via a system of three equations. Performance analysis of the IA estimation methodology for the Student's t distribution demonstrates that the method converges to the maximum likelihood estimate. Closed-form estimates of the location and scale are determined from the mean of the IA pairs and the second moment of the IA triplets, respectively. For the Student's t distribution, the geometric mean of the original samples provides a third equation to determine the shape, though its nonlinear solution requires an iterative solver. With 10,000 samples the relative bias of the parameter estimates is less than 0.01 and the relative precision is less than +/- 0.1. Statistical physics applications are carried out for both a small sample (331) astrophysics dataset and a large sample (2 x 10^8) standard map simulation.