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A compound class of Weibull and power series distributions

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  • Morais, Alice Lemos
  • Barreto-Souza, Wagner

Abstract

In this paper we introduce the Weibull power series (WPS) class of distributions which is obtained by compounding Weibull and power series distributions, where the compounding procedure follows same way that was previously carried out by Adamidis and Loukas (1998). This new class of distributions has as a particular case the two-parameter exponential power series (EPS) class of distributions (Chahkandi and Ganjali, 2009), which contains several lifetime models such as: exponential geometric (Adamidis and Loukas, 1998), exponential Poisson (Kus, 2007) and exponential logarithmic (Tahmasbi and Rezaei, 2008) distributions. The hazard function of our class can be increasing, decreasing and upside down bathtub shaped, among others, while the hazard function of an EPS distribution is only decreasing. We obtain several properties of the WPS distributions such as moments, order statistics, estimation by maximum likelihood and inference for a large sample. Furthermore, the EM algorithm is also used to determine the maximum likelihood estimates of the parameters and we discuss maximum entropy characterizations under suitable constraints. Special distributions are studied in some detail. Applications to two real data sets are given to show the flexibility and potentiality of the new class of distributions.

Suggested Citation

  • Morais, Alice Lemos & Barreto-Souza, Wagner, 2011. "A compound class of Weibull and power series distributions," Computational Statistics & Data Analysis, Elsevier, vol. 55(3), pages 1410-1425, March.
    • Handle: RePEc:eee:csdana:v:55:y:2011:i:3:p:1410-1425
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    1. Silva, Rodrigo B. & Barreto-Souza, Wagner & Cordeiro, Gauss M., 2010. "A new distribution with decreasing, increasing and upside-down bathtub failure rate," Computational Statistics & Data Analysis, Elsevier, vol. 54(4), pages 935-944, April.
    2. Chahkandi, M. & Ganjali, M., 2009. "On some lifetime distributions with decreasing failure rate," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 4433-4440, October.
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    5. Adamidis, K. & Loukas, S., 1998. "A lifetime distribution with decreasing failure rate," Statistics & Probability Letters, Elsevier, vol. 39(1), pages 35-42, July.
    6. Kus, Coskun, 2007. "A new lifetime distribution," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4497-4509, May.
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    Cited by:

    1. Amal S. Hassan & Salwa M. Assar, 2021. "A New Class of Power Function Distribution: Properties and Applications," Annals of Data Science, Springer, vol. 8(2), pages 205-225, June.
    2. Almalki, Saad J. & Nadarajah, Saralees, 2014. "Modifications of the Weibull distribution: A review," Reliability Engineering and System Safety, Elsevier, vol. 124(C), pages 32-55.
    3. Shovan Chowdhury, 2014. "Compounded Generalized Weibull Distributions - A Unified Approach," Working papers 148, Indian Institute of Management Kozhikode.
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    5. Francisco Louzada & M�rcia A.C. Macera & Vicente G. Cancho, 2015. "The Poisson-exponential model for recurrent event data: an application to bowel motility data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(11), pages 2353-2366, November.
    6. Silva, Rodrigo B. & Bourguignon, Marcelo & Dias, Cícero R.B. & Cordeiro, Gauss M., 2013. "The compound class of extended Weibull power series distributions," Computational Statistics & Data Analysis, Elsevier, vol. 58(C), pages 352-367.
    7. Barreto-Souza, Wagner, 2012. "Bivariate gamma-geometric law and its induced Lévy process," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 130-145.
    8. Mojtaba Alizadeh & Seyyed Fazel Bagheri & Mohammad Alizadeh & Saralees Nadarajah, 2017. "A new four-parameter lifetime distribution," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(5), pages 767-797, April.
    9. Bobotas, Panayiotis & Koutras, Markos V., 2019. "Distributions of the minimum and the maximum of a random number of random variables," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 57-64.
    10. Rasool Roozegar & G. G. Hamedani & Leila Amiri & Fatemeh Esfandiyari, 2020. "A New Family of Lifetime Distributions: Theory, Application and Characterizations," Annals of Data Science, Springer, vol. 7(1), pages 109-138, March.
    11. Mahmoudi, Eisa & Jafari, Ali Akbar, 2012. "Generalized exponential–power series distributions," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 4047-4066.
    12. Shovan Chowdhury & Asok K Nanda, 2015. "A special class of distorted premium principle based on an extension of the exponential-geometric distribution," Working papers 188, Indian Institute of Management Kozhikode.
    13. Rasool Roozegar & Saralees Nadarajah & Eisa Mahmoudi, 2022. "The Power Series Exponential Power Series Distributions with Applications to Failure Data Sets," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(1), pages 44-78, May.
    14. Bao Yiqi & Cibele Maria Russo & Vicente G. Cancho & Francisco Louzada, 2016. "Influence diagnostics for the Weibull-Negative-Binomial regression model with cure rate under latent failure causes," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(6), pages 1027-1060, May.
    15. Sanku Dey & Vikas Kumar Sharma & Mhamed Mesfioui, 2017. "A New Extension of Weibull Distribution with Application to Lifetime Data," Annals of Data Science, Springer, vol. 4(1), pages 31-61, March.
    16. Vasileios M. Koutras & Markos V. Koutras, 2020. "Exact Distribution of Random Order Statistics and Applications in Risk Management," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1539-1558, December.
    17. Sandeep Kumar Maurya & Saralees Nadarajah, 2021. "Poisson Generated Family of Distributions: A Review," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 484-540, November.
    18. Broderick O. Oluyede & Boikanyo Makubate & Adeniyi F. Fagbamigbe & Precious Mdlongwa, 2018. "A New Burr XII-Weibull-Logarithmic Distribution for Survival and Lifetime Data Analysis: Model, Theory and Applications," Stats, MDPI, vol. 1(1), pages 1-15, June.
    19. Jimut Bahan Chakrabarty & Shovan Chowdhury, 2016. "Compounded Inverse Weibull Distributions: Properties, Inference and Applications," Working papers 213, Indian Institute of Management Kozhikode.
    20. Saralees Nadarajah & Božidar Popović & Miroslav Ristić, 2013. "Compounding: an R package for computing continuous distributions obtained by compounding a continuous and a discrete distribution," Computational Statistics, Springer, vol. 28(3), pages 977-992, June.
    21. Francesca Condino & Filippo Domma, 2017. "A new distribution function with bounded support: the reflected generalized Topp-Leone power series distribution," METRON, Springer;Sapienza Università di Roma, vol. 75(1), pages 51-68, April.
    22. Vicente G. Cancho & Márcia A. C. Macera & Adriano K. Suzuki & Francisco Louzada & Katherine E. C. Zavaleta, 2020. "A new long-term survival model with dispersion induced by discrete frailty," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 26(2), pages 221-244, April.
    23. Beatriz R. Lanjoni & Edwin M. M. Ortega & Gauss M. Cordeiro, 2016. "Extended Burr XII Regression Models: Theory and Applications," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 21(1), pages 203-224, March.
    24. Bagheri, S.F. & Bahrami Samani, E. & Ganjali, M., 2016. "The generalized modified Weibull power series distribution: Theory and applications," Computational Statistics & Data Analysis, Elsevier, vol. 94(C), pages 136-160.
    25. Muhammad H Tahir & Gauss M. Cordeiro, 2016. "Compounding of distributions: a survey and new generalized classes," Journal of Statistical Distributions and Applications, Springer, vol. 3(1), pages 1-35, December.

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