Brazilian Journal of Probability and Statistics

On the Nielsen distribution

Fredy Castellares, Artur J. Lemonte, and Marcos A. C. Santos

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We introduce a two-parameter discrete distribution that may have a zero vertex and can be useful for modeling overdispersion. The discrete Nielsen distribution generalizes the Fisher logarithmic (i.e., logarithmic series) and Stirling type I distributions in the sense that both can be considered displacements of the Nielsen distribution. We provide a comprehensive account of the structural properties of the new discrete distribution. We also show that the Nielsen distribution is infinitely divisible. We discuss maximum likelihood estimation of the model parameters and provide a simple method to find them numerically. The usefulness of the proposed distribution is illustrated by means of three real data sets to prove its versatility in practical applications.

Article information

Braz. J. Probab. Stat., Volume 34, Number 1 (2020), 90-111.

Received: December 2017
Accepted: August 2018
First available in Project Euclid: 3 February 2020

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Zentralblatt MATH identifier

Discrete distribution Fisher logarithmic distribution Nielsen expansion Stirling distributions


Castellares, Fredy; Lemonte, Artur J.; Santos, Marcos A. C. On the Nielsen distribution. Braz. J. Probab. Stat. 34 (2020), no. 1, 90--111. doi:10.1214/18-BJPS414.

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  • Allison, P. D. (2012). Logistic Regression Using SAS: Theory and Application, 2nd ed. Cary, North Carolina: SAS Institute Inc.
  • Barbiero, A. (2014). An alternative discrete skew Laplace distribution. Statistical Methodology 16, 47–67.
  • Catcheside, D. G., Lea, D. E. and Thoday, J. M. (1946a). Types of chromosome structural change induced by the irradiation of Tradescantia microspores. Journal of Genetics 47, 113–136.
  • Catcheside, D. G., Lea, D. E. and Thoday, J. M. (1946b). The production of chromosome structural changes in Tradescantia microspores in relation to dosage, intensity and temperature. Journal of Genetics 47, 137–149.
  • Englehardt, J. D. and Li, R. C. (2011). The discrete Weibull distribution: An alternative for correlated counts with confirmation for microbial counts in water. Risk Analysis 31, 370–381.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Volume II. New York: John Wiley & Sons.
  • Fisher, R. A., Corbet, A. S. and Williams, C. B. (1943). The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology 12, 42–58.
  • Flajonet, P. and Sedgewick, R. (2009). Analytic Combinatorics. New York: Cambridge University Press.
  • Gómez-Déniz, E., Sarabia, J. M. and Calderin-Ojeda, E. (2011). A new discrete distribution with actuarial applications. Insurance Mathematics & Economics 48, 406–412.
  • Gossiaux, A. and Lemaire, J. (1981). Methodes d’ajustement de distributions de sinistres. Bulletin of the Association of Swiss Actuaries 81, 87–95.
  • Graham, R., Knuth, D. and Patashnik, O. (1994). Concrete Mathematics: A Foundation for Computer Science, 2nd ed. New York: Addison-Wesley.
  • Inusah, S. and Kozubowski, T. J. (2006). A discrete analogue of the Laplace distribution. Journal of Statistical Planning and Inference 136, 1090–1102.
  • Jazi, M. A., Lai, C. D. and Alamatsaz, M. H. (2010). A discrete inverse Weibull distribution and estimation of its parameters. Statistical Methodology 7, 121–132.
  • Johnson, N. L. and Kotz, S. (1982). Developments in discrete distributions, 1969–1980. International Statistical Review 50, 71–101.
  • Jones, M. C. (2015). On families of distributions with shape parameters. International Statistical Review 83, 175–192.
  • Kozubowski, T. J. and Inusah, S. (2006). A skew Laplace distribution on integers. Annals of the Institute of Statistical Mathematics 58, 555–571.
  • Krishna, H. and Pundir, P. S. (2009). Discrete Burr and discrete Pareto distributions. Statistical Methodology 6, 177–188.
  • Nekoukhou, V., Alamatsaz, M. H. and Bidram, H. (2013). Discrete generalized exponential distribution of a second type. Statistics 47, 876–887.
  • Nielsen, N. (1906). Handbuch der Theorie der Gammafunction. Leipzig: Teubner.
  • Nooghabi, M. S., Roknabadi, A. H. R. and Borzadaran, G. M. (2011). Discrete modified Weibull distribution. Metron LXIX, 207–222.
  • Patil, G. P. (1963). Minimum variance unbiased estimation and certain problems of additive number theory. The Annals of Mathematical Statistics 34, 1050–1056.
  • Patil, G. P. and Wani, J. K. (1965). On certain structural properties of the logarithmic series distribution and the first type Stirling distribution. Sankhya Series A 27, 271–280.
  • R Core Team (2016). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
  • Ross, S. M. (2013). Simulation, 5th ed. London: Academic Press.
  • Roy, D. (2004). Discrete Rayleigh distribution. IEEE Transactions on Reliability 53, 255–260.
  • Sichel, H. S. (1951). The estimation of the parameters of a negative binomial distribution with special reference to psychological data. Psychometrika 16, 107–127.
  • Ward, M. (1934). The representation of Stirling’s numbers and Stirling’s polynomials as sums of factorial. American Journal of Mathematics 56, 87–95.