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, [Formula: see text] and [Formula: see text] , of the smallest x and the largest x in N-point samples. A fit of [Formula: see text] or [Formula: see text] 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.