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(N3) operations, where N denotes sample size. Here I develop an approach for estimating the lower and upper bounds that involve O(N) operations. The approach centers on calculating the mean values, x̂min and x̂max, of the smallest x and the largest x in N-point samples. A fit of x̂min or x̂max 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.

中文翻译：

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有界范围内的幂律：根据样本数据估计下限和上限

幂律分布在化学物理学、地球物理学、生物学等领域广泛观察。这些分布的自变量 x 具有强制下界，并且在许多情况下还有上限。从样本数据估计这些界限非常困难，最近的方法涉及 O(N3) 运算，其中 N 表示样本大小。在这里，我开发了一种估计涉及 O(N) 运算的下限和上限的方法。该方法的重点是计算 N 点样本中最小 x 和最大 x 的平均值 x̂min 和 x̂max。将 x̂min 或 x̂max 作为 N 的函数进行拟合可得出下限或上限的估计值。对合成数据的应用证明了该方法的准确性和可靠性。