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We discuss multiple classes of piece-wise Pareto-like power law probabilitydensity functions $p(x)$ with two regimes, a non-pathological core withnon-zero, finite values for support $0\leq x\leq x_{\mathrm{min}}$ and apower-law tail with exponent $-\alpha$ for $x>x_{\mathrm{min}}$. The cores takethe 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 $\beta=0$ leadingto core $p(x)=\mathrm{const}$. We derive explicit maximum-likelihood estimatorsand/or efficient numerical methods to find the best-fit parameter values forempirical data. Solutions for the special cases $\alpha=\beta$ are presented,as well. The results are made available as a Python package.

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