We develop a variation of a Kolmogorov-Smirnov (KS) method for estimating a power law region, including its lower and upper bounds, of the probability density in a set of data which can be modelled as a continuous random sample. Our main innovation is to stabilize the estimation of the bounds of the power law region by introducing an adaptive penalization term involving the logarithmic length of the interval when minimizing the Kolmogorov-Smirnov distance between the random sample and the power law fit over various candidate intervals. We show through simulation studies that an adaptively penalized Kolmogorov-Smirnov (apKS) method improves the estimation of the power law interval on random samples from various theoretical probability distributions. Variability in the estimation of the bounds can be further reduced when the apKS method is applied to subsamples of the original random sample, and the subsample estimates are averaged to yield a final estimate.

我们开发了一种Kolmogorov-Smirnov（KS）方法的变体，用于估计一组可以建模为连续随机样本的数据中的概率密度的幂律区域，包括其上下限。我们的主要创新在于，通过在最小化随机样本与适用于各种候选区间的幂函数之间的Kolmogorov-Smirnov距离时引入包含区间对数长度的自适应惩罚项，来稳定幂律区域的边界估计。我们通过仿真研究表明，自适应惩罚的Kolmogorov-Smirnov（apKS）方法可改善从各种理论概率分布的随机样本上的幂律区间的估计。