Properties and estimation of a bivariate geometric model with locally constant failure rates

  • Alessandro BarbieroEmail author
S.I.: Statistical Reliability Modeling and Optimization


Stochastic models for correlated count data have been attracting a lot of interest in the recent years, due to their many possible applications: for example, in quality control, marketing, insurance, health sciences, and so on. In this paper, we revise a bivariate geometric model, introduced by Roy (J Multivar Anal 46:362–373, 1993), which is very appealing, since it generalizes the univariate concept of constant failure rate—which characterizes the geometric distribution within the class of all discrete random variables—in two dimensions, by introducing the concept of “locally constant” bivariate failure rates. We mainly focus on four aspects of this model that have not been investigated so far: (1) pseudo-random simulation, (2) attainable Pearson’s correlations, (3) stress–strength reliability parameter, and (4) parameter estimation. A Monte Carlo simulation study is carried out in order to assess the performance of the different estimators proposed and application to real data, along with a comparison with alternative bivariate discrete models, is provided as well.


Attainable correlations Correlated counts Failure rate Gumbel–Barnett copula Method of moments Mean residual life Stress–strength model 



I would like to thank the Editor-in-Chief, the Guest Editor, and the anonymous referees for their valuable comments on an earlier draft of this article. I acknowledge the financial support to the present research by the University of Milan (Piano di Sostegno alla Ricerca 2015/2017-Linea 2A).


  1. Barnett, V. (1980). Some bivariate uniform distributions. Communications in Statistics - Theory and Methods, 9(4), 453–461.CrossRefGoogle Scholar
  2. Basu, A. P., & Dhar, S. K. (1995). Bivariate geometric distribution. Journal of Applied Statistical Science, 2, 33–44.Google Scholar
  3. Bracquemond, C, Cretois, E., & Gaudoin, O. (2002). A comparative study of goodness-of-fit tests for the geometric distribution and application to discrete time reliability. Laboratoire Jean Kuntzmann, Applied Mathematics and Computer Science, Technical Report.Google Scholar
  4. Dhar, S. K. (1998). Data analysis with discrete analog of Freund’s model. Journal of Applied Statistical Science, 7, 169–183.Google Scholar
  5. Fiondella, L., & Zeephongsekul, P. (2016). Trivariate Bernoulli distribution with application to software fault tolerance. Annals of Operations Research, 244(1), 241–255.CrossRefGoogle Scholar
  6. Freund, J. E. (1961). A bivariate extension of the exponential distribution. Journal of the American Statistical Association, 56(296), 971–977.CrossRefGoogle Scholar
  7. Galambos, J., & Kotz, S. (1978). Characterizations of probability distributions. Berlin: Springer.CrossRefGoogle Scholar
  8. Gumbel, E. J. (1960). Bivariate exponential distributions. Journal of the American Statistical Association, 55(292), 698–707.CrossRefGoogle Scholar
  9. Hawkes, A. G. (1972). A bivariate exponential distribution with applications to reliability. Journal of the Royal Statistical Society Series B, 34(1), 129–131.Google Scholar
  10. Huber, M., & Maric, N. (2014). Minimum correlation for any bivariate geometric distribution. Alea, 11(1), 459–470.Google Scholar
  11. Jovanović, M. (2017). Estimation of \(P\{ X < Y\} \) for geometric-exponential model based on complete and censored samples. Communications in Statistics - Simulation and Computation, 46(4), 3050–3066.CrossRefGoogle Scholar
  12. Khan, M. S. A., Khalique, A., & Abouammoh, A. M. (1989). On estimating parameters in a discrete Weibull distribution. IEEE Transactions on Reliability, 38(3), 348–350.CrossRefGoogle Scholar
  13. Krishna, H., & Pundir, P. S. (2009). A bivariate geometric distribution with applications to reliability. Communications in Statistics - Theory and Methods, 38(7), 1079–1093.CrossRefGoogle Scholar
  14. Maiti, S. S. (1995). Estimation of \(P (X \le Y)\) in the geometric case. Journal of Indian Statistical Association, 33, 87–91.Google Scholar
  15. Mari, D. D., & Kotz, S. (2001). Correlation and dependence. Singapore: World Scientific.CrossRefGoogle Scholar
  16. Marshall, A. W., & Olkin, I. (1967). A multivariate exponential distribution. Journal of the American Statistical Association, 62(317), 30–44.CrossRefGoogle Scholar
  17. Mitchell, C. R., & Paulson, A. S. (1981). A new bivariate negative binomial distribution. Naval Research Logistic Quarterly, 28(3), 359–374.CrossRefGoogle Scholar
  18. Nelsen, R. B. (1999). An introduction to Copulas. New York: Springer.CrossRefGoogle Scholar
  19. Paulson, A. S., & Uppuluri, V. R. R. (1972). A characterization of the geometric distribution and a bivariate geometric distribution. Sankhyā Series A, 34(3), 297–300.Google Scholar
  20. Phatak, A. G., & Sreehari, M. (1981). Some characterizations of a bivariate geometric distribution. Journal of Indian Statistical Association, 19, 141–146.Google Scholar
  21. R Core Team. (2018). R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria. URL
  22. Roy, D. (1993). Reliability measures in the discrete bivariate set-up and related characterization results for a bivariate geometric distribution. Journal of Multivariate Analysis, 46(2), 362–373.CrossRefGoogle Scholar
  23. Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Université de Paris, 8, 229–231.Google Scholar
  24. Sun, K., & Basu, A. P. (1995). A characterization of a bivariate geometric distribution. Statistics and Probability Letters, 23(4), 307–311.CrossRefGoogle Scholar
  25. Xekalaki, E. (1983). Hazard functions and life distributions in discrete time. Communications in Statistics - Theory and Methods, 12(21), 2503–2509.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Economics, Management and Quantitative MethodsUniversità degli Studi di MilanoMilanItaly

Personalised recommendations