Segmental estimation and testing method for power-law distributions and some extensions

For discrete segmental power-law distributions, the probability ratio ft/ft+1 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.