Main Article Content

Abstract

A new four-parameter lifetime model is introduced and studied. The new model derives its flexibility and wide applicability from the well-known exponentiated Weibull model. Many bivariate and the multivariate type versions are derived using the Morgenstern family and Clayton copula. The new density can exhibit many important shapes with different skewness and kurtosis which can be unimodal and bimodal. The new hazard rate can be decreasing, J-shape, U-shape, constant, increasing, upside down and increasing-constant hazard rates. Various of its structural mathematical properties are derived. Graphical simulations are used in assessing the performance of the estimation method. We proved empirically the importance and flexibility of the new model in modeling various types of data such as failure times, remission times, survival times and strengths data.

Keywords

Marshall-Olkin Family Lehmann Weibull Distribution Order Statistics Maximum Likelihood Estimation Simulation Generating Function Quantile function Moments

Article Details

How to Cite
Shehata, W., & Yousof, H. M. (2021). The four-parameter exponentiated Weibull model with Copula, properties and real data modeling. Pakistan Journal of Statistics and Operation Research, 17(3), 649-667. https://doi.org/10.18187/pjsor.v17i3.3311

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. DOI: https://doi.org/10.6339/JDS.201604_14(2).0004
  2. 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. DOI: https://doi.org/10.18187/pjsor.v11i1.873
  3. Afify, A. Z., Yousof, H. M., Cordeiro, G. M., Ortega, E. M. M. and Nofal, Z. M. (2016b). The Weibull Fréchet distribution and its applications. Journal of Applied Statistics, 43 (14), 2608--2626. DOI: https://doi.org/10.1080/02664763.2016.1142945
  4. Al-Babtain, A. A. Elbatal, I. and Yousof, H. M. (2020a). A new exible three-parameter model: properties, Clayton Copula, and modeling real data. 12(13), 1-26. doi: 10.3390/sym1203044017 DOI: https://doi.org/10.3390/sym12030440
  5. Al-Babtain, A. A. Elbatal, I. and Yousof, H. M. (2020b). A new three parameter Fréchet model with mathematical properties and applications. Journal of Taibah University for Science, 14(1), 265--278. DOI: https://doi.org/10.1080/16583655.2020.1733767
  6. 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. DOI: https://doi.org/10.6339/JDS.201707_15(3).0005
  7. 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. DOI: https://doi.org/10.15672/HJMS.2017.440
  8. Alzaatreh, A., Lee, C. and Famoye, F. (2013). A new method for generating families of continuous distributions, Metron 71, 63--79. DOI: https://doi.org/10.1007/s40300-013-0007-y
  9. Aryal, 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. DOI: https://doi.org/10.5539/ijsp.v6n1p126
  10. Aryal, G. R. and Yousof, H. M. (2017b). The exponentiated generalized-G Poisson family of distributions. Economic Quality Control, 32(1), 1-17. DOI: https://doi.org/10.1515/eqc-2017-0004
  11. Bjerkedal, T. (1960). Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. American Journal of Hygiene, 72, 130--148. DOI: https://doi.org/10.1093/oxfordjournals.aje.a120129
  12. Bourguignon, M., Silva, R.B. and Cordeiro, G.M. (2014). The Weibull--G family of probability distributions, Journal of Data Science 12, 53--68. DOI: https://doi.org/10.6339/JDS.201401_12(1).0004
  13. 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. DOI: https://doi.org/10.1080/00949655.2017.1351972
  14. 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. DOI: https://doi.org/10.1007/s00009-017-0955-1
  15. 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 DOI: https://doi.org/10.1080/02331888.2012.748769
  16. 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 DOI: https://doi.org/10.1016/j.jfranklin.2010.06.010
  17. 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. DOI: https://doi.org/10.1080/00949655.2017.1392524
  18. Elbatal, I. and Aryal, G. (2013). On the transmuted additive Weibull distribution. Austrian Journal of Statistics, 42(2), 117-132. DOI: https://doi.org/10.17713/ajs.v42i2.160
  19. Hamedani, G. G. Yousof, H. M., Rasekhi, M., Alizadeh, M., Najibi, S. M. (2018). Type I general exponential class of distributions. Pak. J. Stat. Oper. Res.,forthcoming DOI: https://doi.org/10.18187/pjsor.v14i1.2193
  20. Hamedani, G. G. Rasekhi, M., Najibi, S. M., Yousof, H. M. and Alizadeh, M. (2018). Type II general exponential class of distributions. Pak. J. Stat. Oper. Res.,forthcoming. DOI: https://doi.org/10.18187/pjsor.v15i2.1699
  21. Khan, M. N. (2015). The modified beta Weibull distribution. Hacettepe Journal of Mathematics and Statistics, 44, 1553--1568.
  22. 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.
  23. Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and Applications of Fractional Difierential Equations. Elsevier, Amsterdam.
  24. Korkmaz, M. C. Yousof, H. M., Rasekhi, M. Hamedani G. G. (2017). The exponential Lindley odd log-logistic G family: properties, characterizations and applications. Journal of Statistical Theory and Applications, forthcoming. DOI: https://doi.org/10.2991/jsta.2018.17.3.10
  25. 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. DOI: https://doi.org/10.22237/jmasm/1177992960
  26. Lehmann, E. L. (1953). The power of rank tests. Annals of Mathematical Statistics 24, 23-43. DOI: https://doi.org/10.1214/aoms/1177729080
  27. Marshall, A. W. and Olkin, I. (1996). A new method for adding a parameter to a family of distributions with application to the exponential and weibull families. Biometrika, 84, 641--652. DOI: https://doi.org/10.1093/biomet/84.3.641
  28. Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42, 299-302. DOI: https://doi.org/10.1109/24.229504
  29. Mudholkar, G. S., Srivastava, D. K. and Freimer, M. (1995). The exponentiated Weibull family: A reanalysis of the bus-motor-failure data. Technometrics, 37, 436-445. DOI: https://doi.org/10.1080/00401706.1995.10484376
  30. Murthy, D. N. P., Xie, M. and Jiang, R. (2004). Weibull Models. John Wiley and Sons, Hoboken, New Jersey.
  31. Nadarajah, S., Cordeiro, G. M. and Ortega, E. M. M. (2013). The exponentiated Weibull distribution: A survey, Statistical Papers, 54, 839-877. DOI: https://doi.org/10.1007/s00362-012-0466-x
  32. Nofal, Z. M., Afify, A. Z., Yousof, H. M. and Cordeiro, G. M. (2017). The generalized transmuted-G family of distributions. Communications in Statistics-Theory and Methods, 46, 4119-4136. DOI: https://doi.org/10.1080/03610926.2015.1078478
  33. Provost, S.B. Saboor, A. and Ahmad, M. (2011). The gamma--Weibull distribution, Pak. Journal Stat., 27, 111--131.
  34. Rezaei, S., Nadarajah, S. and Tahghighnia, N. A (2013). new three-parameter lifetime distribution, Statistics, 47, 835--860. DOI: https://doi.org/10.1080/02331888.2011.627587
  35. Rinne, H. (2009). The Weibull Distribution: A Handbook. CRC Press, Boca Raton, Florida.
  36. Ristic, M.M. and Balakrishnan, N. (2012). The gamma-exponentiated exponential distribution, Journal of Statistical Computation and Simulation, 82, 1191--1206. DOI: https://doi.org/10.1080/00949655.2011.574633
  37. Tian, Y., Tian, M. and Zhu, Q. (2014). Transmuted linear exponential distribution: a new generalization of the linear exponential distribution. Communications in Statistics - Simulation and Computation, 43(10), 2661-2677. DOI: https://doi.org/10.1080/03610918.2013.763978
  38. Weibull, W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech.-Trans, 18(3), 293-297. DOI: https://doi.org/10.1115/1.4010337
  39. Yadav, A.S., Goual, H., Alotaibi, R.M. Rezk, H., 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 di¤erent methods of estimation. Symmetry, 12, 1-26. doi: 10.3390/sym12010057 DOI: https://doi.org/10.3390/sym12010057
  40. Yousof, H. M., Afify, A. Z., Alizadeh, M., Butt, N. S., Hamedani, G. G. and Ali, M. M. (2015). The transmuted exponentiated generalized-G family of distributions. Pak. J. Stat. Oper. Res., 11, 441-464. DOI: https://doi.org/10.18187/pjsor.v11i4.1164
  41. 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.
  42. 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. DOI: https://doi.org/10.2991/jsta.2017.16.4.3
  43. 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. DOI: https://doi.org/10.2991/jsta.2017.16.3.2
  44. 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 DOI: https://doi.org/10.6339/JDS.201710_15(4).00008
  45. 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. DOI: https://doi.org/10.29252/jsri.15.1.45
  46. 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.