Main Article Content

Abstract

A new three-parameter life parametric model called the Marshall-Olkin generalized Weibull is defined and studied. Relevant properties are mathematically derived and analyzed. The new density exhibits various important symmetric and asymmetric shapes with different useful kurtosis. The new failure rate can be “constant”, “upside down-constant (reversed U-HRF-constant)”, “increasing then constant”, “monotonically increasing”, “J-HRF” and “monotonically decreasing”. The method of maximum likelihood is employed to estimate the unknown parameters. A graphical simulation is performed to assess the performance of the maximum likelihood estimation. We checked and proved empirically the importance, applicability and flexibility of the new Weibull model in modeling various symmetric and asymmetric types of data. The new distribution has a high ability to model different symmetric and asymmetric types of data.

Keywords

Marshall-Olkin Family Weibull distribution Order Statistics Maximum Likelihood Estimation Quantile function Generating Function Moments

Article Details

How to Cite
Khalil, M. G., & Kamel, W. M. (2020). The Three-parameters Marshall-Olkin Generalized Weibull Model with Properties and Different Applications to Real Data Sets. Pakistan Journal of Statistics and Operation Research, 16(4), 675-688. https://doi.org/10.18187/pjsor.v16i4.2339

References

  1. Afify, A. Z., Cordeiro, G. M., Yousof, H. M. Alzaatreh, A., and Nofal, Z. M. (2016a). The Kumaraswamy transmuted-G family of distributions: properties and applications, 14, 245-270.
  2. Afify, A.Z., Cordeiro, G. M., Yousof, H. M., Saboor, A. and Ortega, E. M. M. (2016b). The Marshall-Olkin additive Weibull distribution with variable shapes for the hazard rate. Hacettepe Journal of Mathematics and Statistics, 47, 365-381.
  3. Afify, A. Z., Nofal, Z. M. and Ebraheim, A. N. (2015). Exponentiated transmuted generalized Rayleigh distribution: A new four parameter Rayleigh distribution. Pak.j.stat.oper.res, 11 (1), 115-134.
  4. Afify, A. Z., Yousof, H. M., Cordeiro, G. M., Ortega, E. M. M. and Nofal, Z. M. (2016c). The Weibull Fréchet distribution and its applications. Journal of Applied Statistics, 43 (14), 2608–2626.
  5. Al-Babtain, A. A., Elbatal, I. and Yousof, H. M. (2020a). A new three parameter Fréchet model with mathematical properties and applications. Journal of Taibah University for Science, 14(1), 265-278.‏
  6. Al-babtain, A. A., Elbatal, I. and Yousof, H. M. (2020b). A new flexible three-parameter model: properties, clayton copula, and modeling real data. Symmetry, 12(3), 440.‏
  7. Alizadeh, M., Ghosh, I., Yousof, H. M., Rasekhi, M. and Hamedani G. G. (2017). The generalized odd generalized exponential family of distributions: properties, characterizations and applications, J. Data Sci. 15, 443-466.
  8. Alizadeh, M., Jamal, F., Yousof, H. M., Khanahmadi, M. and Hamedani, G. G. (2020a). Flexible Weibull generated family of distributions: characterizations, mathematical properties and applications. University Politehnica of Bucharest Scientific Bulletin-series A - Applied Mathematics and Physics, 82(1), 145-150.‏
  9. Alizadeh, M., Rasekhi, M., Yousof, H. M. and Hamedani G. G. (2018). The transmuted Weibull G family of distributions. Hacettepe Journal of Mathematics and Statistics, 47(6), 1-20.
  10. Alizadeh, M., Yousof, H. M., Jahanshahi, S. M. A., Najibi, S. M. and Hamedani, G. G. (2020b). The transmuted odd log-logistic-G family of distributions. Journal of Statistics and Management Systems, 1-27.‏
  11. Arya, G. R., Ortega, E. M., Hamedani, G. G. and Yousof, H. M. (2017a). The ToppLeone Generated Weibull distribution: regression model, characterizations and applications, International Journal of Statistics and Probability, 6, 126-141.
  12. Aryal, G. R. and Yousof, H. M. (2017b). The exponentiated generalized-G Poisson family of distributions. Economic Quality Control, 32(1), 1-17.
  13. Bjerkedal, T. (1960). Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. American Journal of Hygiene, 72, 130--148.
  14. Bourguignon, M., Silva, R.B. and Cordeiro, G.M. (2014). The Weibull-G family of probability distributions, Journal of Data Science 12, 53--68.
  15. Brito, E., Cordeiro, G. M., Yousof, H. M., Alizadeh, M. and Silva, G. O. (2017). Topp-Leone Odd Log-Logistic Family of Distributions, Journal of Statistical Computation and Simulation, 87(15), 3040--3058.
  16. Cordeiro, G. M., Afify, A. Z., Yousof, H. M., Pescim, R. R. and Aryal, G. R. (2017a). The exponentiated Weibull-H family of distributions: Theory and Applications. Mediterranean Journal of Mathematics, 14, 1-22.
  17. Cordeiro, G.M., Afify, A. Z., Yousof, H. M., Cakmakyapan, S. and Ozel, G. (2018). The Lindley Weibull distribution: properties and applications, Anais da Academia Brasileira de Ciências, forthcoming.
  18. Cordeiro, G. M., Hashimoto, E. M., Edwin, E. M. M. Ortega. (2014). The McDonald Weibull model. Statistics: A Journal of Theoretical and Applied Statistics, 48, 256—278.
  19. Cordeiro, G. M., Ortega, E. M. and Nadarajah, S. (2010). The Kumaraswamy Weibull distribution with application to failure data. Journal of the Franklin Institute, 347, 1399--1429.
  20. Cordeiro, G. M., Yousof, H. M., Ramires, T. G. and Ortega, E. M. M. (2017b). The Burr XII system of densities: properties, regression model and applications. Journal of Statistical Computation and Simulation, 88(3), 432-456.
  21. Elbatal, I. and Aryal, G. (2013). On the transmuted additive Weibull distribution. Austrian Journal of Statistics, 42(2), 117-132.
  22. Goual, H., Yousof, H. M. and Ali, M. M. (2019a). Validation of the odd Lindley exponentiated exponential by a modified goodness of fit test with applications to censored and complete data. Pak. J. Stat. Oper. Res. 15(3), 745-771.
  23. Goual, H. and Yousof, H. M. (2019b). Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodness-of-fit test. Journal of Applied Statistics, 47(1), 1-32.
  24. Karamikabir, H., Afshari, M., Yousof, H. M., Alizadeh, M. and Hamedani, G. (2020). The Weibull Topp-Leone Generated Family of Distributions: Statistical Properties and Applications. Journal of The Iranian Statistical Society, 19(1), 121-161.‏
  25. Khalil, M. G., Hamedani G. G. and Yousof, H. M. (2019). The Burr X exponentiated Weibull model: characterizations, mathematical properties and applications to failure and survival times data. Pak. J. Stat. Oper. Res., XV(1), 141-160.
  26. Khan, M. N. (2015). The modified beta Weibull distribution. Hacettepe Journal of Mathematics and Statistics, 44, 1553-1568.
  27. Khan, M. S. and King, R. (2013). Transmuted modified Weibull distribution: a generalization of the modified Weibull probability distribution. European Journal of Pure and Applied Mathematics, 6, 66-88.
  28. Korkmaz, M. C., Alizadeh, M., Yousof, H. M. and Butt, N. S. (2018a). The generalized odd Weibull generated family of distributions: statistical properties and applications. Pak. J. Stat. Oper. Res., 14(3), 541-556.
  29. Korkmaz, M. Ç., Altun, E., Yousof, H. M. and Hamedani, G. G. (2020). The Hjorth's IDB Generator of Distributions: Properties, Characterizations, Regression Modeling and Applications. Journal of Statistical Theory and Applications, 19(1), 59-74.‏
  30. Korkmaz, M. Ç., Yousof, H. M. and Hamedani, G. G. (2018b). The exponential Lindley odd log-logistic-G family: properties, characterizations and applications. Journal of Statistical Theory and Applications, 17(3), 554-571.‏
  31. Korkmaz, M. C., Yousof, H. M., Hamedani G. G. and Ali, M. M. (2018c). The Marshall–Olkin generalized G Poisson family of distributions, Pakistan Journal of Statistics, 34(3), 251-267.
  32. Korkmaz, M. C., Yousof, H. M., Rasekhi, M. and Hamedani, G. G. (2018d). The Odd Lindley Burr XII Model: Bayesian Analysis, Classical Inference and Characterizations. Journal of Data Science, 16(2), 327-353.‏
  33. Lee, E. T. and Wang, J. (2003). Statistical methods for survival data analysis (Vol. 476). John Wiley & Sons.‏
  34. Lee, C., Famoye, F. and Olumolade, O. (2007). Beta Weibull distribution: some properties and applications to censored data. Journal of Modern Applied Statistical Methods, 6, 17.
  35. Merovci, F., Alizadeh, M., Yousof, H. M. and Hamedani G. G. (2017). The exponentiated transmuted-G family of distributions: theory and applications, Communications in Statistics-Theory and Methods, 46(21), 10800-10822.
  36. Merovci, F., Yousof, H. and Hamedani, G. G. (2020). The Poisson Topp Leone Generator of Distributions for Lifetime Data: Theory, Characterizations and Applications. Pakistan Journal of Statistics and Operation Research, 16(2), 343-355.‏
  37. Mansour, M. M., Ibrahim, M., Aidi, K., Shafique Butt, N., Ali, M. M., Yousof, H. M. and Hamed, M. S. (2020a). A New Log-Logistic Lifetime Model with Mathematical Properties, Copula, Modified Goodness-of-Fit test for Validation and Real Data Modeling. Mathematics, 8(9), 1508.
  38. Mansour, M. M., Butt, N. S., Ansari, S. I., Yousof, H. M., Ali, M. M., Ibrahim, M. (2020b). A new exponentiated Weibull distribution’s extension: copula, mathematical properties and applications. Contributions to Mathematics, 1 (2020) 57–66. DOI: 10.47443/cm.2020.0018
  39. Mansour, M., Korkmaz, M. C, Ali, M. M., Yousof, H. M., Ansari, S. I. and Ibrahim, M. (2020c). A Generalization of the exponentiated Weibull model with properties, copula and application. Eurasian Bulletin of Mathematics, 3(2), 84-102.
  40. Mansour, M., Rasekhi, M., Ibrahim, M., Aidi, K., Yousof, H. M. and Elrazik, E. A. (2020d). A New Parametric Life Distribution with Modified Bagdonavičius-Nikulin Goodness-of-Fit Test for Censored Validation, Properties, Applications, and Different Estimation Methods. Entropy, 22(5), 592.
  41. Mansour, M., Yousof, H. M., Shehata, W. A. and Ibrahim, M. (2020e). A new two parameter Burr XII distribution: properties, copula, different estimation methods and modeling acute bone cancer data. Journal of Nonlinear Science and Applications, 13(5), 223-238.
  42. Mansour, M. M., Butt, N. S., Yousof, H., Ansari, S. I. and Ibrahim, M. (2020f). A Generalization of Reciprocal Exponential Model: Clayton Copula, Statistical Properties and Modeling Skewed and Symmetric Real Data Sets. Pakistan Journal of Statistics and Operation Research, 373-386.
  43. Nadarajah, S., Cordeiro, G. M. and Ortega, E. M. M. (2015). The exponentiated Weibull distribution: A survey, Statistical Papers, 54, 839-877.
  44. Provost, S.B. Saboor, A. and Ahmad, M. (2011). The gamma Weibull distribution, Pak. Journal Stat., 27, 111--131.
  45. Ristic, M.M. and Balakrishnan, N. (2012). The gamma-exponentiated exponential distribution, Journal of Statistical Computation and Simulation, 82, 1191-1206.
  46. Weibull, W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech.-Trans, 18(3), 293-297.
  47. Yadav, A. S., Goual, H., Alotaibi, R. M., Ali, M. M. and Yousof, H. M. (2020). Validation of the Topp-Leone-Lomax model via a modified Nikulin-Rao-Robson goodness-of-fit test with different methods of estimation. Symmetry, 12(1), 57.‏
  48. Yousof, H. M., Afify, A. Z., Alizadeh, M., Nadarajah, S., Aryal, G. R. and Hamedani, G. G. (2018a). The Marshall-Olkin generalized-G family of distributions with Applications, STATISTICA, 78(3), 273- 295.
  49. Yousof, H. M., Afify, A. Z., Cordeiro, G. M., Alzaatreh, A., and Ahsanullah, M. (2017a). A new four-parameter Weibull model for lifetime data. Journal of Statistical Theory and Applications, 16(4), 448 - 466.
  50. Yousof, H. M., Afify, A. Z., Hamedani, G. G. and Aryal, G. (2017b). The Burr X generator of distributions for lifetime data. Journal of Statistical Theory and Applications, 16, 288--305.
  51. Yousof, H. M., Alizadeh, M., Jahanshahi, S. M. A., Ramires, T. G., Ghosh, I. and Hamedani G. G. (2017c). The transmuted Topp-Leone G family of distributions: theory, characterizations and applications, Journal of Data Science. 15, 723-740
  52. Yousof, H. M., Majumder, M., Jahanshahi, S. M. A., Ali, M. M. and Hamedani G. G. (2018b). A new Weibull class of distributions: theory, characterizations and applications, Journal of Statistical Research of Iran, 15, 45-83.
  53. Yousof, H. M., Rasekhi, M., Afify, A. Z., Alizadeh, M., Ghosh, I. and Hamedani G. G. (2017d). The beta Weibull-G family of distributions: theory, characterizations and applications, Pakistan Journal of Statistics, 33, 95-116.