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

Benjamin F. Maier

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arXiv · 2309.09589

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

Benjamin F. Maier

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Abstract

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 $0\leq x\leq x_{\mathrm{min}}$ and a power-law tail with exponent $-α$ for $x>x_{\mathrm{min}}$. The cores take the respective shapes (i) $p(x)\propto (x/x_{\mathrm{min}})^β$, (ii) $p(x)\propto\exp(-β[x/x_{\mathrm{min}}-1])$, and (iii) $p(x)\propto [2-(x/x_{\mathrm{min}})^β]$, including the special case $β=0$ leading to core $p(x)=\mathrm{const}$. 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.

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Benjamin F. Maier. 2023-09-18. Maximum-likelihood fits of piece-wise Pareto distributions with finite and non-zero core. https://arxiv.org/abs/2309.09589

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