Journal of the Iranian Statistical Society

A Discrete Kumaraswamy Marshall-Olkin Exponential Distribution

Authors

1 Department of Statistics, CHRIST (Deemed to be University), Hosur Road, Bangalore, Karnataka, 560029, India.

2 Department of Statistics, Deva Matha College, Kuravilangad, Kerala, 686633, India.

3 Department of Statistics, The Islamia University of Bahawalpur, Punjab 63100, Pakistan.

4 Universite de Caen, LMNO, Campus II, Science 3, 14032, Caen, France.

10.52547/jirss.20.2.129

Abstract

Finding new families of distributions has become a popular tool in statistical research. In this article, we introduce a new flexible four-parameter discrete model based on the Marshall-Olkin approach, namely, the discrete Kumaraswamy Marshall-Olkin exponential distribution. The proposed distribution can be viewed as another generalization of the geometric distribution and enfolds some important distributions as special cases. Some properties of the new distribution are derived. The model parameters are estimated by the maximum likelihood method, with validation through a complete simulation study. The usefulness of the new model is illustrated via count-type real data sets.

Keywords

  1. Aarset, M.V. (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability. 36:106-108. Akinsete, A., Famoye, F. and Lee, C. (2014) The Kumaraswamy-geometric distribution. Journal of Statistical Distributions and Applications. 1:1--21. Alizadeh, M. Tahir, M.H., Cordeiro, G.M., Zubair, M. and Hamedani, G.G. (2015). The Kumaraswamy Marshall-Olkin family of distributions. Journal of the Egyptian Mathematical Society, 23:546--557. Alzaatreh, A., Lee, C. and Famoye, F. (2012). On the discrete analogues of continuous distributions. Statistical Methods. 9:589--603. Amigo, J.M., Balogh, S.G. and Hernandez, S. (2018). A brief review of generalized entropies, Entropy, 20, 813. Chakraborty, S. (2015). A new discrete distribution related to generalized gamma distribution and its properties. Communication in Statistics-Theory and Methods. 44:1691--1705. Chakraborty, S. and Chakravarty, D. (2016). A new discrete probability distribution with integer support on ($-infty,infty $). Communication in Statistics-Theory and Methods. 45:492--505. Cordeiro, G.M. and de Castro, M. (2011). A new family of generalized distributions, Journal of Statistics Computation and Simulation 81:883--893. Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics, London: Chapman & Hall. G'{o}mez-D'{e}niz, E. (2010). Another generalization of the geometric distribution. Test.19:399--415. Gupta, R.C. and Gupta, R.D. (2007). Proportional reversed hazard rate model and its applications. Journal of Statistics and Planning Inference. 137:3525--3536. Gupta, R.C., Gupta, P.L. and Gupta, R.D. (1998). Modeling failure time data by Lehman alternatives. Communications in Statistics-Theory Methods. 27:887--904. Jayakumar, K. and Thomas, M. (2008). On a generalization of Marshall-Olkin scheme and its application to Burr type XII distribution. Statistical Papers. 49:421--439. Jayakumar, K. and Sankaran, K.K. (2017a). A Discrete Generalization of Marshall-Olkin Scheme and its Application to Geometric Distribution. Journal of the Kerala Statistical Association, 28:1--21. Jayakumar, K. and Sankaran, K.K (2017b). A generalization of discrete Weibull distribution, Communications in Statistics-Simulation and Computation. 47:24, 6064--6078. Krishna, H. and Pundir, P.S. (2009). Discrete Burr and discrete Pareto distributions. Statistical Methodology. 6:177--188. Lisman, J.H.C. and van Zuylen, M.C.A. (1972). Note on the generation of the most probable frequency distribution. Statistica Neerlandica. 26:19--23. Marshall, A.W. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrica. 84:641--652. Murthy, D.N.P., Xie, M. and Jiang, R. (2004). Weibull models. New Jersey: JohnWiley and Sons. Nakagawa, T. and Osaki, S. (1975). The discrete Weibull distribution. IEEE Transactions on Reliability. 24:300--301. Sato, H., Ikota, M., Aritoshi, S. and Masuda, H. (1999). A new defect distribution meteorology with a consistent discrete exponential formula and its applications. IEEE Transactions on Semiconductor Manufacturing. 12:409--418. Stein, W.E. and Dattero, R. (1984). A new discrete Weibull distribution. IEEE Transactions on Reliability. 33:196--197. Steutel, F.W. and van Harn, K. (2004). Infinite Divisibility of Probability Distributions on the Real Line. New York: Marcel Dekker. Supanekar, S.R. and Shirke, D.T. (2015). A new discrete family of distributions. ProbStat Forum. 8:83--94. Tahir, M.H. and Nadarajah, S. (2015). Parameter induction in continuous univariate distributions:Well established G families, Annals of the Brazilian Academy of Sciences. 87: 539--568. Xie, M and, Goh, T.N. (1993). Improvement detection by control charts for high yield processes. International Journal of Quality & Reliability Management. 10:24--31.
  2. Aarset, M. V. (1987), How to identify a bathtub hazard rate. IEEE Transactions on Reliability , 36(1), 106-108. [DOI:10.1109/TR.1987.5222310]
  3. Akinsete, A., Famoye, F., and Lee, C. (2014), The Kumaraswamy-geometric distribution. Journal of Statistical Distributions and Applications, 1, 1-21. [DOI:10.1186/s40488-014-0017-1]
  4. Alizadeh, M. Tahir, M. H., Cordeiro, G. M., Zubair, M., and Hamedani, G. G. (2015), The Kumaraswamy Marshall-Olkin family of distributions. Journal of the Egyptian Mathematical Society , 23(3), 546-557. [DOI:10.1016/j.joems.2014.12.002]
  5. Alzaatreh, A., Lee, C., and Famoye, F. (2012), On the discrete analogues of continuous distributions. Statistical Methods , 9(6), 589-603. [DOI:10.1016/j.stamet.2012.03.003]
  6. Amigo, J. M., Balogh, S. G., and Hernandez, S. (2018), A brief review of generalized entropies. Entropy , 20, 813. [DOI:10.3390/e20110813]
  7. Chakraborty, S. (2015), A new discrete distribution related to generalized gamma distribution and its properties. Communication in Statistics-Theory and Methods , 44(8), 1691-1705. [DOI:10.1080/03610926.2013.781635]
  8. Chakraborty, S., and Chakravarty, D. (2016), A new discrete probability distribution with integer support on (-∞,∞). Communication in Statistics-Theory and Method , 45(2), 492-505. [DOI:10.1080/03610926.2013.830743]
  9. Cordeiro, G. M., and de Castro, M. (2011), A new family of generalized distributions. Journal of Statistics Computation and Simulation , 81(7), 883-893. [DOI:10.1080/00949650903530745]
  10. Cox, D. R., and Hinkley, D. V. (1974), Theoretical Statistics, London: Chapman & Hall. [DOI:10.1007/978-1-4899-2887-0]
  11. Gomez-Deniz, E. (2010), Another generalization of the geometric distribution. Test , 19(2), 399-415. [DOI:10.1007/s11749-009-0169-3]
  12. Gupta, R. C., and Gupta, R. D. (2007), Proportional reversed hazard rate model and its applications. Journal of Statistics and Planning Inference , 137(11), 3525-3536. [DOI:10.1016/j.jspi.2007.03.029]
  13. Gupta, R. C., Gupta, P. L., and Gupta, R. D. (1998), Modeling failure time data by Lehman alternatives. Communications in Statistics-Theory Methods , 27(4), 887-904. [DOI:10.1080/03610929808832134]
  14. Jayakumar, K., and Thomas, M. (2008), On a generalization of Marshall-Olkin scheme and its application to Burr type XII distribution. Statistical Papers , 49(3), 421-439. [DOI:10.1007/s00362-006-0024-5]
  15. Jayakumar, K., and Sankaran, K. K. (2017a), A discrete generalization of Marshall-Olkin scheme and its application to geometric distribution. Journal of the Kerala Statistical Association , 28, 1-21.
  16. Jayakumar, K., and Sankaran, K. K. (2017b), A generalization of discrete Weibull distribution. Communications in Statistics-Simulation and Computation , 47(24), 6064-6078. [DOI:10.1080/03610926.2017.1406115]
  17. Krishna, H., and Pundir, P. S. (2009), Discrete Burr and discrete Pareto distributions. Statistical Methodology , 6(2), 177-188. [DOI:10.1016/j.stamet.2008.07.001]
  18. Lisman, J. H. C., and van Zuylen, M. C. A. (1972), Note on the generation of the most probable frequency distribution. Statistica Neerlandica , 26(1), 19-23. [DOI:10.1111/j.1467-9574.1972.tb00152.x]
  19. Marshall, A. W., and Olkin, I. (1997), A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrica , 84(3), 641-652. [DOI:10.1093/biomet/84.3.641]
  20. Murthy, D. N. P., Xie, M., and Jiang, R. (2004), Weibull models. New Jersey: John Wiley and Sons.
  21. Nakagawa, T., and Osaki, S. (1975), The discrete Weibull distribution. IEEE Transactions on Reliability , 24(5), 300-301. [DOI:10.1109/TR.1975.5214915]
  22. Sato, H., Ikota, M., Aritoshi, S., and Masuda, H. (1999), A new defect distribution meteorology with a consistent discrete exponential formula and its applications. IEEE Transactions on Semiconductor Manufacturing , 12(4), 409-418. [DOI:10.1109/66.806118]
  23. Stein, W. E., and Dattero, R. (1984), A new discrete Weibull distribution. IEEE Transactions on Reliability , 33(2), 196-197. [DOI:10.1109/TR.1984.5221777]
  24. Steutel, F. W., and van Harn, K. (2004), Infinite Divisibility of Probability Distributions on the Real Line. New York: Marcel Dekker. [DOI:10.1201/9780203014127]
  25. Supanekar, S. R., and Shirke, D. T. (2015), A new discrete family of distributions. ProbStat Forum , 8, 83-94.
  26. Tahir, M. H., and Nadarajah, S. (2015), Parameter induction in continuous univariate distributions: Well established G families. Annals of the Brazilian Academy of Sciences , 87(2), 539-568. [DOI:10.1590/0001-3765201520140299]
  27. Xie, M., and Goh, T. N. (1993), Improvement detection by control charts for high yield processes. International Journal of Quality & Reliability Management , 10(7), 24-31. [DOI:10.1108/02656719310043779]
Journal of the Iranian Statistical Society
Volume 20, Issue 2
December 2021
Pages 129-152
  • Receive Date: 23 July 2022
  • Revise Date: 19 May 2024
  • Accept Date: 23 July 2022