Bayesian Estimates for the Parameters of a Power-law Distribution with Exponential Cutoff using Monte Carlo Methods
$$ \gdef{\boldphi}{\boldsymbol{\phi}} \gdef{\xmin}{x_{\text{min}}} \gdef{\gammaone} {\Gamma_{\alpha}^{\prime} (1-\alpha, \lambda \xmin)} \gdef{\gammatwo}{\Gamma_{\alpha}^{\prime \prime} (1-\alpha, \lambda \xmin)} \gdef{\gammazero}{\Gamma (1-\alpha, \lambda \xmin)} $$ Introduction The power-law distribution is of the form \( f(x) \propto x^{-\alpha} \), where \( \alpha \) is called the scaling parameter. It models many natural phenomena like acoustic attenuation, Curie–Von Schweidler law, neuronal avalanches, and others. As \( x \to 0 \), \( f(x) \) diverges out of bounds. Hence, we gdefine \( \xmin \) as a lower bound for the support of the distribution function \( f(x) \). Using this, the power-law density function can be written as ...