  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 the utilization of a modified group of moments
for estimating a heavy-tailed distribution. These modified moments are
determined from powers of the original distribution. 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. A least absolute deviation criteria
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.
