A novel goodness of fit test for the truncated and non-truncated Yule distributions

The Yule distribution has a wide range of applications. However whether empirical data in some application fields can be considered to follow a Yule distribution has always been a controversial topic. As a heavy-tailed distribution, Yule distributions have a typical feature: the ratio of the probability at the point k to the probability at the point k+1 is a linear function of the parameter. Based on this property this paper proposes a goodness of fit test for Yule distributions. The most typical improvement of the proposed testing method is that it does not rely on parameter estimation, and it is applicable to both truncated and non-truncated Yule distributions. Simulation results demonstrate the applicability of the proposed tests and its advantages compared to Pearson chi-square test from the perspective of Pitman asymptotic relative efficiency. We also obtain a consistent estimate and an asymptotic confidence interval of the parameter. The simulation results show the accuracy of this estimation method and its advantages compared to maximum likelihood estimation. Finally, two examples of real data analysis demonstrate that the proposed testing method can explore better fitting distributions for empirical data.