Independent Approximates enable closed-form estimation of heavy-tailed distributions
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Independent Approximates (IAs) are proven to enable a closed-form estimation of heavy-tailed distributions with an analytical density such as the generalized Pareto and Student's t distributions. A broader proof using convolution of the characteristic function is described for future research. (IAs) are selected from independent, identically distributed samples by partitioning the samples into groups of size n and retaining the median of the samples in those groups which have approximately equal samples. The marginal distribution along the diagonal of equal values has a density proportional to the nth power of the original density. This nth power density, which the IAs approximate, has faster tail decay enabling closed-form estimation of its moments and retains a functional relationship with the original density. Computational experiments with between 1000 to 100,000 Student's t samples are reported for over a range of location, scale, and shape (inverse of degree of freedom) parameters. IA pairs are used to estimate the location, IA triplets for the scale, and the geometric mean of the original samples for the shape. With 10,000 samples the relative bias of the parameter estimates is less than 0.01 and a relative precision is less than plus or minus 0.1. The theoretical bias is zero for the location and the finite bias for the scale can be subtracted out. The theoretical precision has a finite range when the shape is less than 2 for the location estimate and less than 3/2 for the scale estimate. The boundary of finite precision can be extended using higher-order IAs.
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