  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.
