Computational Statistics

, Volume 33, Issue 2, pp 807–836 | Cite as

On the zero-modified Poisson–Shanker regression model and its application to fetal deaths notification data

  • Wesley BertoliEmail author
  • Katiane S. Conceição
  • Marinho G. Andrade
  • Francisco Louzada
Original Paper


In this paper, we propose the zero-modified Poisson–Shanker regression model as an alternative to model overdispersed count data exhibiting inflation or deflation of zeros in the presence of covariates. The zero modification has been incorporated using the zero-truncated Poisson–Shanker distribution. The zero-modified Poisson–Shanker distribution has been written as a hurdle model using a simple reparameterization of the probability function which leads to the fact that the proposed model can be fitted without any previous information about the zero modification present in a given dataset. The standard Bayesian procedures have been considered for estimation and inference. A simulation study has been presented to illustrate the performance of the developed methodology. The usefulness of the proposed model has been evaluated using a real dataset on fetal deaths notification data in Bahia State, Brazil. A sensitivity study to detect points which can influence the parameter estimates has been performed using Kullback–Leibler divergence measure. The randomized quantile residuals have been considered for the model validation issue. General comparison of the proposed model with some well-known discrete distributions has been provided.


Poisson–Shanker distribution Zero inflated/deflated data Zero-modified and hurdle models Bayesian estimation Influential points 



The authors are grateful for the insightful comments and constructive suggestions provided by the associate editor and the anonymous referees. Also, the first author would like to thank the Federal Technology University of Paraná and the Araucária Foundation for the financial support during this research.

Supplementary material

180_2017_788_MOESM1_ESM.txt (8 kb)
Supplementary material 1 (txt 8 KB)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Wesley Bertoli
    • 1
    • 2
    Email author
  • Katiane S. Conceição
    • 3
  • Marinho G. Andrade
    • 3
  • Francisco Louzada
    • 3
  1. 1.Departmento Acadêmico de MatemáticaUniversidade Tecnológica Federal do ParanáCuritibaBrazil
  2. 2.Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São Paulo, São CarlosSão PauloBrazil
  3. 3.Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São PauloSão CarlosBrazil

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