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

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 $-\alpha$ for $x>x_{\mathrm{min}}$. The cores take the respective shapes (i) $p(x)\propto (x/x_{\mathrm{min}})^\beta$, (ii) $p(x)\propto\exp(-\beta[x/x_{\mathrm{min}}-1])$, and (iii) $p(x)\propto [2-(x/x_{\mathrm{min}})^\beta]$, including the special case $\beta=0$ 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 $\alpha=\beta$ are presented, as well. The results are made available as a Python package.