The Discrete Inverse Burr Distribution with Characterizations, Properties, Applications, Bayesian and Non-Bayesian Estimations

  • Christophe Chesneau University of Caen, France
  • Haitham Yousof Benha university, Egypt
  • G.G. Hamedani Department of Mathematical and Statistical Sciences, Marquette University, USA
  • Mohamed Ibrahim Damietta University, Damietta, Egypt
Keywords: discretization, characterization, weighted least square estimation, maximum likelihood estimation, count data, Bayesian estimation, ordinary least square estimation


A new one-parameter heavy tailed discrete distribution with infinite mean is defined and studied. The probability mass function of the new distribution can be "unimodal and right skewed" and its failure rate can be monotonically decreasing. Some of its relevant properties are discussed. Some characterizations based on: (i) the conditional expectation of a certain function of the random variable and (ii) in terms of the reversed hazard function are presented. Different Bayesian and non-Bayesian estimation methods are described and compared using simulations and two real data applications are given. The new model is used to model carious teeth data and counts of cysts in kidneys datasets, and it outperforms many well-known competitive discrete models.

Author Biography

Christophe Chesneau, University of Caen, France
LMNO, Mathematics department


M. Aboraya, M. H. Yousof, G. G. Hamedani, and M. Ibrahim. A new family of discrete distributions with mathematical properties, characterizations, Bayesian and non-Bayesian estimation methods, Mathematics, vol. 8, no. 10, 1648, 2020.

M. Bebbington, C. D. Lai, M. Wellington, and R. Zitikis. The discrete additive Weibull distribution: A bathtub-shaped hazard for discontinuous failure data, Reliability Engineering & System Safety, vol. 106, pp. 37–44, 2012.

W. Bodhisuwan, and S. Sangpoom. The discrete weighted Lindley distribution, In Proceedings of the International Conference on Mathematics, Statistics, and Their Applications, Banda Aceh, Indonesia, 4–6 October 2016, 2016.

L. Cai. Metropolis-Hastings Robbins-Monro algorithm for confirmatory item factor analysis, Journal of Educational and Behavioral Statistics, vol. 35, no. 3, pp. 307–335, 2010.

G. Casella, and R. L. Berger, Statistical Inference, Brooks/Cole Publishing Company, California, 1990.

M. T. Chao, The asymptotic behavior of Bayes’ estimators, The Annals of Mathematical Statistics, vol. 41, no. 2, pp. 601–608, 1970.

S. Chib, and E. Greenberg. Understanding the metropolis-hastings algorithm, The american statistician, vol. 49, no. 4, pp. 327–335, 1995.

S. Chan, P. R. Riley, K. L. Price, F. McElduff, and P. J. Winyard. Corticosteroid-induced kidney dysmorphogenesis is associated with deregulated expression of known cystogenic molecules, as well as Indian hedgehog, The American Journal of Physiology - Renal Physiology, vol. 298, pp. 346–356, 2009.

M. El-Morshedy, M. S. Eliwa, and H. Nagy. A new two-parameter exponentiated discrete Lindley distribution: Properties, estimation and applications, Journal of Applied Statistics, vol. 47, pp. 354–375, 2020.

M. El-Morshedy, M. S. Eliwa, and E. Altun. Discrete Burr-Hatke distribution with properties, estimation methods and regression model, IEEE Access, vol. 8, pp. 74359–74370, 2020.

E. Gomez-D´eniz. Another generalization of the geometric distribution, Test, vol. 19, no. 2, pp. 399–415, 2010.

E. Gomez-D´eniz, and E. Caldern-Ojeda. The discrete Lindley distribution: Properties and applications, Journal of Statistical Computation and Simulation, vol. 81, no. 11, pp. 1405–1416, 2011.

T. Hussain, and M. Ahmad. Discrete inverse Rayleigh distribution, Pakistan Journal of Statistics, vol. 30, no. 2, pp. 203–222, 2014.

T. Hussain, M. Aslam, and M. A. Ahmad. Two parameter discrete Lindley distribution, Revista Colombiana de Estad´ıstica, vol. 39, no. 1, pp. 45–61, 2016.

M. Ibrahim, M. M. Ali, and H. M. Yousof, The discrete analogue of the Weibull G family: properties, different applications, Bayesian and non-Bayesian estimation methods, Annals of Data Science, forthcoming, 2021.

I. A. Ibragimov. Some limit theorems for stationary processes, Theory of Probability & Its Applications, vol. 7, no. 4, pp. 349-382, 1962.

A. M. Jazi, D. C. Lai, and H. M. Alamatsaz. Inverse Weibull distribution and estimation of its parameters, Statistical Methodology, vol. 7, no. 2, pp. 121–132, 2010.

A. W. Kemp. Classes of discrete lifetime distributions, Communications in Statistics - Theory and Methods, vol. 33, no. 12, pp. 3069–3093, 2004.

A. W. Kemp. The discrete half-normal distribution, In: Advances in mathematical and statistical modeling, Birkhauser, Basel, pp. 353–365, 2008.

M. C. Korkmaz, H. M. Yousof, M. Rasekhi, and G. G. Hamedani. The Odd Lindley Burr XII Model: Bayesian Analysis, Classical Inference and Characterizations, Journal of Data Science, vol. 16, no. 2, pp. 327–353, 2018.

H. Krishna, and P. S. Pundir. Discrete Burr and discrete Pareto distributions, Statistical Methodology, vol. 6, no. 2, pp. 177–188, 2009.

C. Kumar, Y. M. Tripathi, and M. K. Rastogi. On a discrete analogue of linear failure rate distribution American journal of

mathematical and management sciences, vol. 36, no. 3, pp. 229–246, 2017.

T. Nakagawa, and S. Osaki. The discrete Weibull distribution, IEEE Transactions on Reliability, vol. 24, no. 5, pp. 300–301, 1975.

V. Nekoukhou, M. H. Alamatsaz, and H. Bidram. Discrete generalized exponential distribution of a second type, Statistics, vol. 47, no. 4, pp. 876–887, 2013.

B. A. Para, and T. R. Jan. Discrete version of log-logistic distribution and its applications in genetics, International Journal of Modern Mathematical Sciences, vol. 14, pp. 407–422, 2016.

B. A. Para, and T. R. Jan. On discrete three-parameter Burr type XII and discrete Lomax distributions and their applications to model count data from medical science, Biometrics & Biostatistics International Journal, vol. 4, pp. 1–15, 2016.

S. D. Poisson. Probabilit´e des Jugements en Mati´ere Criminelle et en Mati´ere Civile, Pr´ec´ed´ees des R´eGles G´enerales du Calcul des Probabiliti´es, Bachelier: Paris, France, pp. 206–207, 1837.

D. Roy. Discrete Rayleigh distribution, IEEE Transactions on Reliability, vol. 53, pp. 255–260, 2004.

M. Sankaran. The discrete poisson-lindley distribution, Biometrics, vol. 26, no. 1, 145-149, 1970.

W. E. Stein, and R. A. Dattero. A new discrete Weibull distribution, IEEE Transactions on Reliability, vol. R-33, no. 2, pp. 196–197, 1984.

H. M. Yousof, C. Chesneau, G. G. Hamedani, and M. Ibrahim. A new discrete distribution: properties, characterizations, modeling real count data, Bayesian and non-Bayesian estimations, Statistica, vol. 81, no. 2, pp. 135-162, 2021.

How to Cite
Chesneau, C., Yousof, H., Hamedani, G., & Ibrahim, M. (2022). The Discrete Inverse Burr Distribution with Characterizations, Properties, Applications, Bayesian and Non-Bayesian Estimations. Statistics, Optimization & Information Computing, 10(2), 352-371.
Research Articles