Power Law in a Bounded Range: Estimating the Lower and Upper Bounds from Sample Data

Power law distributions are widely observed in chemical physics, geophysics, biology, and beyond. The independent variable x of these distributions has an obligatory lower bound and in many cases also an upper bound. Estimating these bounds from sample data is notoriously difficult, with a recent method involving O(N^3) operations, where N denotes sample size. Here I develop an approach for estimating the lower and upper bounds that involves O(N) operations. The approach centers on calculating the mean values, x_min and x_max, of the smallest x and the largest x in N-point samples. A fit of x_min or x_max as a function of N yields the estimate for the lower or upper bound. Application to synthetic data demonstrates the accuracy and reliability of this approach.