  Power law-like size distributions are ubiquitous in astrophysical
instabilities. There are at least four natural effects that cause deviations
from ideal power law size distributions, which we model here in a generalized
way: (1) a physical threshold of an instability; (2) incomplete sampling of the
smallest events below a threshold $x_0$; (3) contamination by an
event-unrelated background $x_b$; and (4) truncation effects at the largest
events due to a finite system size. These effects can be modeled in simplest
terms with a "thresholded power law" distribution function (also called
generalized Pareto [type II] or Lomax distribution), $N(x) dx \propto
(x+x_0)^{-a} dx$, where $x_0 > 0$ is positive for a threshold effect, while
$x_0 < 0$ is negative for background contamination. We analytically derive the
functional shape of this thresholded power law distribution function from an
exponential-growth evolution model, which produces avalanches only when a
disturbance exceeds a critical threshold $x_0$. We apply the thresholded power
law distribution function to terrestrial, solar (HXRBS, BATSE, RHESSI), and
stellar flare (Kepler) data sets. We find that the thresholded power law model
provides an adequate fit to most of the observed data. Major advantages of this
model are the automated choice of the power law fitting range, diagnostics of
background contamination, physical inastability thresholds, instrumental
detection thresholds, and finite system size limits. When testing
self-organized criticality models, which predict ideal power laws, we suggest
to include these natural truncation effects.
