The Waring distribution is an important two-parameter discrete distribution, commonly used in fields such as ecology, linguistics, and information science, where heavy tails are often observed. In this paper, we propose a new goodness-of-fit test for the Waring distribution, which is established through the hazard rate and a linear equivalent definition of the Waring distribution. We establish an asymptotic Chi-square null distribution for the proposed test and show that it is more powerful than classical methods in simulation studies. Finally, we apply the test to analyze the authorships of published papers on computer science.

References

• Banerjee, B., & Pradhan, B. (2018). Kolmogorov–Smirnov test for life test data with hybrid censoring. Communications in Statistics-Theory and Methods, 47(open in a new window)(11(open in a new window)), 2590–2604. https://doi.org/10.1080/03610926.2016.1205616 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Barlow, R. E., Marshall, A. W., & Proschan, F. (1963). Properties of probability distributions with monotone hazard rate. Annals of Mathematical Statistics, 34(open in a new window)(2(open in a new window)), 375–389. https://doi.org/10.1214/aoms/1177704147 
  • View
   (Open in a new window)Google Scholar
• Chang, S., Li, D., & Qi, Y. (2023). Pearson's goodness-of-fit tests for sparse distributions. Journal of Applied Statistics, 50(open in a new window)(5(open in a new window)), 1078–1093. https://doi.org/10.1080/02664763.2021.2017413 
  • View
   (Open in a new window)PubMed (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Clauset, A., Shalizi, C., & Newman, M. (2009). Power-law distributions in empirical data. SIAM Review, 51(open in a new window)(4(open in a new window)), 661–703. https://doi.org/10.1137/070710111 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Cueva-López, V., Olmo-Jiménez, M. J., & Rodríguez-Avi, J. (2019). EM algorithm for an extension of the Waring distribution. Computational and Mathematical Methods, 1(5), e1046. 
  • View
   (Open in a new window)Google Scholar
• Cueva-López, V., Olmo-Jiménez, M. J., & Rodríguez-Avi, J. (2021). An over and underdispersed biparametric extension of the Waring distribution. Mathematics, 9(open in a new window)(2(open in a new window)), 170. https://doi.org/10.3390/math9020170 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Goldstein, M., Morris, S., & Yen, G. (2004). Problems with fitting to the power-law distribution. The European Physical Journal B, 41(open in a new window)(2(open in a new window)), 255–258. https://doi.org/10.1140/epjb/e2004-00316-5 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Haberman, S. J. (1988). A warning on the use of Chi-squared statistics with frequency tables with small expected cell counts. Journal of the American Statistical Association, 83(open in a new window)(402(open in a new window)), 555–560. https://doi.org/10.1080/01621459.1988.10478632 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Huete-Morales, M. D., & Marmolejo-Martín, J. A. (2020). The Waring distribution as a low-frequency prediction model: A study of organic livestock farms in Andalusia. Mathematics, 8(open in a new window)(11(open in a new window)), 2025. https://doi.org/10.3390/math8112025 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Kang, L., Xu, W., & Jiang, L. (2007). Waring distribution and the application of its parameter estimation method (in Chinese). Statistics and Decision-making, 9, 138–139. 

   (Open in a new window)Google Scholar

• Lilliefors, H. (1967). On the Kolmogorov-Smirnov test for normality with mean and variance unknown. Journal of the American Statistical Association, 62(open in a new window)(318(open in a new window)), 399–402. https://doi.org/10.1080/01621459.1967.10482916 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Lilliefors, H. (1969). On the Kolmogorov-Smirnov test for the exponential distribution with mean unknown. Journal of the American Statistical Association, 64(open in a new window), 387–389. https://doi.org/10.1080/01621459.1969.10500983 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Okamura, H., & Dohi, T. (2019). On Kolmogorov-Smirnov test for software reliability models with grouped data. In 2019 IEEE 19th International Conference On Software Quality, Reliability And Security (QRS), Sofia, Bulgaria (pp. 77–82). IEEE. 
  • View
   (Open in a new window)Google Scholar
• Otsu, T., & Taniguchi, G. (2020). Kolmogorov–Smirnov type test for generated variables. Economics Letters, 195(open in a new window), 109401. https://doi.org/10.1016/j.econlet.2020.109401 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Price, D. (1965). Network of scientific papers. Science, 149(open in a new window)(3683(open in a new window)), 510–515. https://doi.org/10.1126/science.149.3683.510 
  • View
   (Open in a new window)PubMed (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Price, D. (1976). A general theory of bibliometric and other cumulative advantage processes. Journal of the American Society for Information Science, 27(open in a new window)(5(open in a new window)), 292–306. https://doi.org/10.1002/asi.v27:5 
  • View
   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Rempała, G. A., & Wesołowski, J. (2016). Double asymptotics for the Chi-square statistic. Statistics & Probability Letters, 119(open in a new window), 317–325. https://doi.org/10.1016/j.spl.2016.09.004 
  • View
   (Open in a new window)PubMed (Open in a new window)Web of Science ®(Open in a new window)Google Scholar
• Rivas, L., & Campos, F. (2021). Zero inflated Waring distribution. Communications in Statistics – Simulation and Computation, 52, 1–16. 

   (Open in a new window)Web of Science ®(Open in a new window)Google Scholar

• Tang, Y., Wang, J., & Zhu, Z. (2023). On the MLE of the Waring distribution. Statistical Theory and Related Fields, 7(open in a new window)(2(open in a new window)), 144–158. https://doi.org/10.1080/24754269.2023.2176608 
  • View
   (Open in a new window)Google Scholar
• Wang, J., Wu, X., & Xi, H. (2023). A novel test on yule distributions [Tech. Rep.]. East China Normal University. 

   (Open in a new window)Google Scholar

To cite this article: Yanlin Tang, Jinglong Wang, Menghan Yi & Zhongyi Zhu (04 Sep 2024): Goodness of fit for the Waring distribution, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2024.2389758

To link to this article: https://doi.org/10.1080/24754269.2024.2389758