Some new statistical methods for a class of zero-truncated discrete distributions with applications

  • Guo-Liang Tian
  • Xiqian Ding
  • Yin LiuEmail author
  • Man-Lai Tang
Original paper


Counting data without zero category often occurs in various fields. A class of zero-truncated discrete distributions such as the zero-truncated Poisson, zero-truncated binomial and zero-truncated negative-binomial distributions are proposed in literature to model such count data. In this paper, three main contributions have been made for better studying the zero-truncated discrete distributions: First, a novel unified expectation–maximization (EM) algorithm is developed for calculating the maximum likelihood estimates (MLEs) of parameters in general zero-truncated discrete distributions and an important feature of the proposed EM algorithm is that the latent variables and the observed variables are independent, which is unusual in general EM-type algorithms; Second, for those who do not understand the principle of latent variables, a unified minorization–maximization algorithm, as an alternative to the EM algorithm, for obtaining the MLEs of parameters in a class of zero-truncated discrete distributions is discussed; Third, a unified method is proposed to derive the distribution of the sum of discrete random variables, which has important applications in the construction of the shortest Clopper–Pearson confidence intervals of parameters of interest and in the calculation of the exact p value of a two-sided test for small sample sizes in one sample problem.


EM algorithm MM algorithm Shortest confidence intervals Stochastic representation Zero-truncated discrete models 



Yin Liu’s research was fully supported by Grants (Nos. 11601524 and 61773401) from the National Natural Science Foundation of China and a Grant (31721811206) from the Young Teachers Innovation Project of Zhongnan University of Economics and Law. Guo-Liang Tian’s research was fully supported by a Grant from the National Natural Science Foundation of China (No. 11771199). The work of Man-Lai Tang was supported by the Research Fund of the Project (Grant No.UGC/FDS14/P06/17) and National Natural Science Foundation of China (Grant No. 11871124).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsSouthern University of Science and TechnologyShenzhenPeople’s Republic of China
  2. 2.Department of Statistics and Actuarial ScienceThe University of Hong KongLung Fu ShanPeople’s Republic of China
  3. 3.School of Statistics and MathematicsZhongnan University of Economics and LawWuhanPeople’s Republic of China
  4. 4.Department of Mathematics and Statistics, School of Decision SciencesThe Hang Seng University of Hong KongShatin, New TownPeople’s Republic of China

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