Maximum-likelihood fits of piece-wise Pareto distributions with finite and non-zero core

09/18/2023

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by   Benjamin F. Maier, et al.

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We discuss multiple classes of piece-wise Pareto-like power law probability density functions p(x) with two regimes, a non-pathological core with non-zero, finite values for support 0≤ x≤ x_min and a power-law tail with exponent -α for x>x_min. The cores take the respective shapes (i) p(x)∝ (x/x_min)^β, (ii) p(x)∝exp(-β[x/x_min-1]), and (iii) p(x)∝ [2-(x/x_min)^β], including the special case β=0 leading to core p(x)=const. We derive explicit maximum-likelihood estimators and/or efficient numerical methods to find the best-fit parameter values for empirical data. Solutions for the special cases α=β are presented, as well. The results are made available as a Python package.