For discrete segmental power-law distributions, the probability ratio is a linear function of the exponent parameter. Based on this property, the estimation of the exponent parameter and a goodness-of-fit testing method are provided. The proposed testing method is parameter-independent and the testing statistic is proved to asymptotically follow a chi-square distribution. In the region where power-law properties exist, the testing method and the estimation method can be applied by segments, so they can also be used to determine interval endpoints. These methods can also be extended to other discrete distributions such as Yule distribution, Poisson distribution, Geometry distribution and so on. Some simulation results of synthetic truncated power-law distributions provide support for the effectiveness of the proposed methods. To demonstrate the applicability of the method, two empirical examples, word frequencies of the novel Moby Dick and US casuality numbers in the American Indian War, are analyzed.

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幂律分布的分段估计和检验方法及其扩展

对于离散分段幂律分布，概率比是指数参数的线性函数。基于这一性质，提出了指数参数的估计和拟合优度检验方法。所提出的检验方法是参数无关的，并且检验统计量被证明渐近遵循卡方分布。在存在幂律性质的区域，检验方法和估计方法可以分段应用，因此也可以用来确定区间端点。这些方法还可以扩展到其他离散分布，如尤尔分布、泊松分布、几何分布等。合成截断幂律分布的一些仿真结果为所提出方法的有效性提供了支持。为了证明该方法的适用性，我们对小说《白鲸》的词频和美国印第安人战争中的伤亡数字这两个实证例子进行了分析。