Kybernetika 50 no. 1, 142-169, 2014

A new family of compound lifetime distributions

A. Asgharzadeh, Hassan S. Bakouch, Saralees Nadarajah and L. EsmaeiliDOI: 10.14736/kyb-2014-1-0142


In this paper, we introduce a general family of continuous lifetime distributions by compounding any continuous distribution and the Poisson-Lindley distribution. It is more flexible than several recently introduced lifetime distributions. The failure rate functions of our family can be increasing, decreasing, bathtub shaped and unimodal shaped. Several properties of this family are investigated including shape characteristics of the probability density, moments, order statistics, (reversed) residual lifetime moments, conditional moments and Rényi entropy. The parameters are estimated by the maximum likelihood method and the Fisher's information matrix is determined. Several special cases of this family are studied in some detail. An application to a real data set illustrates the performance of the family of distributions.


estimation, moments, Poisson-Lindley distribution, failure rate shapes


62E15, 62E20


  1. K. Adamidis and S. Loukas: A lifetime distribution with decreasing failure rate. Statist. Probab. Lett. 39 (1998), 35-42.   CrossRef
  2. H. S. Bakouch, M. M. Ristic, A. Asgharzadeh, L. Esmaily and B. M. Al-Zahrani: An exponentiated exponential binomial distribution with application. Statist. Probab. Lett. 82 (2012), 1067-1081.   CrossRef
  3. W. Barreto-Souza and H. S. Bakouch: A new lifetime model with decreasing failure rate. Statistics 47 (2013), 465-476.   CrossRef
  4. W. Barreto-Souza, A. L. de Morais and G. M. Cordeiro: The Weibull-geometric distribution. J. Statist. Comput. Simul. 81 (2011), 645-657.   CrossRef
  5. M. Chahkandi and M. Ganjali: On some lifetime distributions with decreasing failure rate. Comput. Statist. Data Anal. 53 (2009), 4433-4440.   CrossRef
  6. M. E. Ghitany, D. K. Al-Mutairi and S. Nadarajah: Zero-truncated Poisson-Lindley distribution and its application. Math. Comput. Simul. 79 (2008), 279-287.   CrossRef
  7. P. L. Gupta and R. C. Gupta: On the moments of residual life in reliability and some characterization results. Comm. Statist.-Theory and Methods 12 (1983), 449-461.   CrossRef
  8. J. R. M. Hosking: L-moments: Analysis and estimation of distributions using linear combinations of order statistics. J. Royal Statist. Soc. B 52 (1990), 105-124.   CrossRef
  9. C. Kus: A new lifetime distribution. Comp. Statist. Data Anal. 51 (2007), 4497-4509.   CrossRef
  10. W. Lu and D. Shi: A new compounding life distribution: The Weibull-Poisson distribution. J. Appl. Statist. 39 (2012), 21-38.   CrossRef
  11. A. J. McNeil: Estimating the tails of loss severity distributions using extreme value theory. Astin Bull. 27 (1997), 117-137.   CrossRef
  12. A. L. Morais and W. Barreto-Souza: A compound class of Weibull and power series distributions. Comput. Statist. Data Anal. 55 (2011), 1410-1425.   CrossRef
  13. G. S. Mudholkar and D. K. Srivastava: Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Trans. Reliability 42 (1993), 299-302.   CrossRef
  14. A. Rényi: On measures of entropy and information. In: Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. I (1961), University of California Press, Berkeley, pp. 547-561.   CrossRef
  15. R. Tahmasbi and S. Rezaei: A two-parameter lifetime distribution with decreasing failure rate. Comput. Statist. Data Anal. 52 (2008), 3889-3901.   CrossRef