A New Generalized Family of Lifetime Distributions Motivated by Parallel and Series Structures

  • Mehdi Goldoust Department of Mathematics, Behbahan Branch, Islamic Azad University, Behbahan, Iran
  • Adel Mohammadpour Department of Statistics, Faculty of Mathematics & Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
  • Morad Alizadeh Department of Statistics, Persian Gulf University, Bushehr, Iran
  • Gholamhosein Hamedani Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, USA
Keywords: Reliability, Power series and Compound distributions, Truncated moments.


Any given system can be represented as a parallel arrangement of series structures. Motivated by this fact, a general family of distributions is introduced, by adding two extra parameters to a distribution (called baseline distribution),twice compounding with power series distribution. The new family can allow various hazard rate curves that compete well with other alternatives in fitting real data. We derive formal expressions for its moments, generating function, mean residual lifetime and other reliability functions. Certain characterizations of the new family are presented in terms of the ratio of two truncated moments as well as based on the hazard rate function. The maximum likelihood estimation technique is used to estimate the model parameters and a simulation study is conducted to investigate the performance of the maximum likelihood estimates. Finally, two applications of the model with real data sets are presented to illustrate the usefulness of the proposed distribution.

Author Biography

Morad Alizadeh, Department of Statistics, Persian Gulf University, Bushehr, Iran
Department of Statistics, Persian Gulf University, Bushehr, Iran


M. V. Aarset, How to identify bathtub hazard rate, IEEE Transactions on Reliability, Vol. 36, pp. 106–108, 1987.

I.Abdul-Moniem,and H.F.Abdel-Hameed, On exponentiated Lomax distribution, International Journal of Mathematical Education,vol. 33, pp. 1–7, 2012.

M. Alizadeh, M. Emadi, and Mahdi Doostparast, A New Two Parameter Lifetime Distribution: Properties, Applications and Different Method of Estimations, Statistics, Optimization & Information Computing, Vol. 7, pp. 291–310, 2019.

K. Adamidis, T. Dimitrakopoulou, and S. Loukas, On a generalization of the exponential geometric distribution, Statistics and Probability Letters, vol. 73, pp. 259–269, 2005.

K. Adamidis, and S. Loukas, A lifetime distribution with decreasing failure rate, Statistics and Probability Letters, vol. 39, pp.35–42, 1998.

H. M. Barakat, and Y. H. Abdelkader, Computing the moments of order statistics from nonidentical random variables, Statistical Methods and Applications, vol. 13, pp. 15–26, 2004.

W. Barreto-Souza, A. Morais, and G. M. Cordeiro, The Weibull-geometric distribution, Journal of Statistical Computation and Simulation, vol. 81, pp. 645–657, 2011.

W. Barreto-Souza, A. H. S. Santos, and G. M. Cordeiro, The beta generalized exponential distribution, Journal of Statistical Computation and Simulation, vol. 80, pp. 159–172, 2010.

M. Chahkandi, and M. Ganjali, On some lifetime distributions with decreasing failure rate, Computational Statistics and Data Analysis, vol. 53, pp. 4433–4440, 2009.

K. Cooray, Generalization of the Weibull distribution: the odd Weibull family, Statistical Modelling, vol. 6, pp. 265–277, 2006.

K. Cooray, and M. A. Ananda,A Generalization of the Half-Normal Distribution with Applications to Lifetime Data,Communications in Statistics - Theory and Methods, vol. 37, no. 9, pp. 1323–1337, 2008.

D. Cox, and D. Hinkley, Theoretical Statistics, Chapman and Hall, New York, 1979.

S. J. Eichhorn, and G.R. Davies, Modelling the crystalline deformation of native and regenerated cellulose, Cellulose, vol. 13,no. 3, pp. 291–307, 2005.

F. Famoye, C. Lee, and O. Olumolade, The beta-Weibull distribution, Journal of Statistical Theory and Applications, vol. 4, pp. 122–136, 2005.

J. Flores, P. Borges, V. G. Cancho, and F. Louzada, The complementary exponential power series distribution, Brazilian Journal of Probability and Statistics, vol. 27, pp. 565–584, 2013.

W. Glänzel, A characterization theorem based on truncated moments and its application to some distribution families, Mathematical Statistics and Probability Theory, vol. 2, pp. 75–84, 1987.

W. Glänzel, Some consequences of a characterization theorem based on truncated moments, Statistics: A Journal of Theoretical and Applied Statistics, vol. 21, no. 4, pp. 613–618, 1990.

I.S. Gradshteyn, and I.M. Ryzhik, Table of Integrals, Series, and Products, Sixth edition, Academic Press, San Diego, 2000.

J. A. Greenwood, J. M. Landwehr, and N. C. Matalas, Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form, Water Resources Research, vol 15, pp. 1049C-1054, 1979.

A. H. Khan, and T.R. Jan, The inverse Weibull-geometric distribution, International Journal of Modern Mathematical Sciences, vol.14, pp. 134–146, 2016.

A. W. Marshall, and I. Olkin, A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families, Biometrika, vol. 84, pp. 641–652, 1997.

A. L. Morais, and W. Barreto-Souza, A compound class of Weibull and power series distributions, Computational Statistics and Data Analysis, vol. 55, pp. 1410–1425, 2011.

S. Nadarajah, and S. Kotz, The beta exponential distribution, Reliability Engineering & System Safety, vol. 91, pp. 689–697, 2006.

A. Noack, A class of random variables with discrete distributions, Annals of Mathematical Statistics, vol. 21, pp. 127–132, 1950.

H. Pham, Handbook of Reliability Engineering. First edition, Springer, London, 2003.

F. Proschan, Theoretical explanation of observed decreasing failure rate, Technometrics, vol. 5, pp. 375–383, 1963.

F. Qi, X. T. Shi, and F. F. Liu, Expansions of the exponential and the logarithm of power series and applications, Arabian Journal of Mathematics, vol. 6, no. 2, pp. 95–108, 2017.

S. M. Ross, Introduction to Probability Models, Tenth edition, Academic Press, Boston, 2010.

C. E. Shannon, A mathematical theory of communication, The Bell System Technical Journal, vol. 27, pp. 379–423, 1948.

R. L. Smith, and J. C. Naylor, A comparison of maximum likelihood and Bayesian estimatorsfor the three-parameter Weibull distribution, Applied Statistics, vol. 36, pp. 358–369, 1987.

R. Tahmasbi, and S. Rezaei, A two-parameter lifetime distribution with decreasing failure rate, Computational Statistics and Data Analysis, vol. 52, pp. 3889–3901, 2008.

S. Tahmasebi, and A. A. Jafari, Exponentiated extended Weibull power series class of distributions, Ciencia e Natura, vol. 37, pp.183–194, 2015.

How to Cite
Goldoust, M., Mohammadpour, A., Alizadeh, M., & Hamedani, G. (2019). A New Generalized Family of Lifetime Distributions Motivated by Parallel and Series Structures. Statistics, Optimization & Information Computing, 7(4), 779-801. https://doi.org/10.19139/soic-2310-5070-719
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