Main Article Content

Abstract

In this article, we introduce a new generalization of the Exponentiated Exponential distribution. Various structural mathematical properties are derived. Numerical analysis for mean, variance, skewness and kurtosis and the dispersion index are performed. The new density can be right skewed and symmetric with "unimodal" and "bimodal" shapes. The new hazard function can be "constant", "monotonically decreasing", " monotonically increasing", "increasing-constantâ€, “upside-down-constant", "decreasing-constant". Many bivariate and multivariate type model have been also derived. We assess the performance of the maximum likelihood method graphically via the biases and mean squared errors. The applicability of the new life distribution is illustrated by means of two real data sets.

Keywords

Exponentiated Exponential Morgenstern family Clayton Copula Real Data Modeling Hazard Function

Article Details

How to Cite
Shehata, W. A. M., Butt, N. S., Yousof, H., & Aboraya, M. (2022). A New Lifetime Parametric Model for the Survival and Relief Times with Copulas and Properties. Pakistan Journal of Statistics and Operation Research, 18(1), 249-272. https://doi.org/10.18187/pjsor.v18i1.3930

References

  1. Aboraya, M. (2019a). A New Extension of the Lomax Distribution with Properties and Applications to Failure Times Data. Pakistan Journal of Statistics and Operation Research (15)2, 461-479.
  2. Aboraya, M. (2019b). The extended generalized inverse Weibull distribution and its applications, Journal of Statistics and Applications, 2(1), 1-17.
  3. Aboraya, M. (2021a). A new one-parameter G family of compound distributions: copulas, statistical properties and applications. Statistics, Optimization & Information Computing, forthcoming.
  4. Aboraya, M. and Butt, N. S. (2019). Extended Weibull Burr XII Distribution: Properties and Applications. Pakistan Journal of Statistics and Operation Research, (15)4, 891-903.
  5. Aboraya, M. (2021b). Marshall-Olkin Lehmann Lomax Distribution: Theory, Statistical Properties, Copulas and Real Data Modeling. Pakistan Journal of Statistics and Operation Research, 17(2), 509-530.
  6. Aidi, K., Butt, N. S., Ali, M. M., Ibrahim, M., Yousof, H. M. and Shehata, W. A. M. (2021). A Modified Chi-square Type Test Statistic for the Double Burr X Model with Applications to Right Censored Medical and Reliability Data. Pakistan Journal of Statistics and Operation Research, 17(3), 615-623.
  7. Al-Babtain, A. A. Elbatal, I. and Yousof, H. M. (2020). A new flexible three-parameter model: properties, Clayton Copula, and modeling real data, Symmetry, 12, 1-17. doi:10.3390/sym12030440
  8. 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.
  9. Aryal, G. R., Ortega, E. M., Hamedani, G. G. and Yousof, H. M. (2017). The Topp Leone Generated Weibull distribution: regression model, characterizations and applications, International Journal of Statistics and Probability, 6, 126-141.
  10. Aslam, M., Kundu, D., Ahmad, M. (2010). Time truncated acceptance sampling plans for generalized exponential distribution. J. Appl. Stat. 37, 555-566.
  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.
  12. 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.
  13. Chao, M. T. (1970). The asymptotic behavior of Bayes' estimators. The Annals of Mathematical Statistics, 41(2), 601-608.
  14. Cordeiro, G. M., Alizadeh, M., Ozel, G., Hosseini, B., Ortega, E. M. M. and Altun, E. (2017). The generalized odd log-logistic family of distributions: properties, regression models and applications. Journal of Statistical Computation andSimulation, 87(5), 908-932.
  15. Cordeiro, G. M., Yousof, H. M., Ramires, T. G. and Ortega, E. M. M. (2018). The Burr XII system of densities: properties, regression model and applications. Journal of Statistical Computation and Simulation, 88(3), 432-456.
  16. Eugene, N., Lee, C., Famoye, F. (2002). Beta-normal distribution and its applications. Communications in Statistics-Theory and Methods 31(4), 497-512.
  17. Farlie, D. J. G. (1960) The performance of some correlation coefficients for a general
  18. bivariate distribution. Biometrika, 47, 307-323.
  19. Ghosh, I. and Ray. S. (2016). Some alternative bivariate Kumaraswamy type distributions via copula with application in risk management. Journal of Statistical Theory and Practice 10,693-706.
  20. Goual, H., Yousof, H. M. and Ali, M. M. (2019). Validation of the odd Lindley exponentiated exponential by a modified goodness of fit test with applications to censored and complete data. Pakistan Journal of Statistics and Operation Research, 15(3), 745-771.â€
  21. Goual, H. and Yousof, H. M. (2020). Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodness-of-fit test. Journal of Applied Statistics, 47(3), 393-423.â€
  22. Goual, H., Yousof, H. M. and Ali, M. M. (2020). Lomax inverse Weibull model: properties, applications, and a modified Chi-squared goodness-of-fit test for validation. Journal of Nonlinear Sciences & Applications (JNSA), 13(6), 330-353.
  23. Gumbel, E. J. (1961). Bivariate logistic distributions. Journal of the American Statistical Association, 56(294), 335-349.â€
  24. Gumbel, E. J. (1960) Bivariate exponential distributions. Journ. Amer. Statist.
  25. Assoc., 55, 698-707.
  26. Gross, J. and Clark, V. A. (1975). Survival Distributions: Reliability Applications in the Biometrical Sciences, John Wiley, New York, USA.
  27. Gupta, R. C. (1981) On the mean residual life function in survival studies, Distributions in Scientific Work 5, D-Reidel Publishing Co. Boston, 327-334.
  28. Gupta, R. D. and Kundu, D. (2001a). Exponentiated exponential family: an alternative to gamma and Weibull distributions. Biom. J. 43, 117—130.
  29. Gupta, R. D. and Kundu, D. (2001b). Generalized exponential distributions: different methods of estimation. J. Stat. Comput. Simul. 69, 315-338.
  30. Gupta, R. D., Kundu, D. (2007). Generalized exponential distribution: existing results and some recent developments. J. Stat. Plan. Inference 137, 3537-3547.
  31. Ibrahim, M., Aidi, K., Ali, M. M. and Yousof, H. M. (2021). A Novel Test Statistic for Right Censored Validity under a new Chen extension with Applications in Reliability and Medicine. Annals of Data Science, forthcoming.
  32. Ibrahim, M., Yadav, A. S., Yousof, H. M., Goual, H. and Hamedani, G. G. (2019). A new extension of Lindley distribution: modified validation test, characterizations and different methods of estimation. Communications for Statistical Applications and Methods, 26(5), 473-495.
  33. Ibragimov, I. A. (1962). Some limit theorems for stationary processes. Theory of Probability & Its Applications, 7(4), 349-382.
  34. Ibrahim, M., Altun, E. and Yousof, H. M. (2020). A new distribution for modeling lifetime data with different methods of estimation and censored regression modeling. Statistics, Optimization and Information Computing, forthcoming.
  35. Johnson, N. L. and Kotz, S. (1975) On some generalized Farlie- Gumbel- Morgenstern distributions. Commun. Stat. Theory, 4, 415-427.
  36. Johnson, N. L. and Kotz, S. (1977) On some generalized Farlie- Gumbel- Morgenstern distributions- II: Regression, correlation and further generalizations. Commun.
  37. Stat.Theory, 6, 485-496.
  38. Korkmaz, M. C., Alizadeh, M., Yousof, H. M. and Butt, N. S. (2018). The generalized odd Weibull generated family of distributions: statistical properties and applications. Pak. J. Stat. Oper. Res., 14(3), 541-556.
  39. Kundu, D., Nanda, A.K., (2010). Some reliability properties of the inactivity time. Communications in Statistics-Theory and Methods 39, 899- 911.
  40. Kundu, D. and Pradhan, B. (2009). Bayesian inference and life testing plans for generalized exponential distribution. Sci. China Ser. A, Math. 52, 1373-1388.
  41. 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.â€
  42. Mansour, M. M., Butt, N. S., Ansari, S. I., Yousof, H. M., Ali, M. M. and 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
  43. Mansour, M., Korkmaz, M. Ç., 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.â€
  44. Mansour, M., Rasekhi, M., Ibrahim, M., Aidi, K., Yousof, H. M. and Elrazik, E. A. (2020d). A New Parametric Life Distribution with Modified BagdonaviÄius-Nikulin Goodness-of-Fit Test for Censored Validation, Properties, Applications, and Different Estimation Methods. Entropy, 22(5), 592.â€
  45. Mansour, M., Yousof, H. M., Shehata, W. A. M. 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.
  46. Mansour, M. M., Butt, N. S., Yousof, H. M., 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, 16(2), 373-386.â€
  47. Marciano, F. W. P., Nascimento, A. D. C., Santos-Neto, M., and Cordeiro, G. M. (2012). The Mc- Γ distribution and its statistical properties: An application to reliability data. International Journal of Statistics and Probability, 1(1), 53-71.
  48. Morgenstern, D. (1956). Einfache beispiele zweidimensionaler verteilungen. Mitteilingsblatt fur Mathematische Statistik, 8, 234-235.
  49. Pougaza, D. B. and Djafari, M. A. (2011). Maximum entropies copulas. Proceedings of the 30th international workshop on Bayesian inference and maximum Entropy methods in Science and Engineering, 329-336.
  50. Rodriguez-Lallena, J. A. and Ubeda-Flores, M. (2004). A new class of bivariate copulas. Statistics and Probability Letters, 66, 315-25.
  51. 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.â€
  52. Yousof, H. M., Aidi, K., Hamedani, G. G and Ibrahim, M. (2021a). A new parametric lifetime distribution with modified Chi-square type test for right censored validation, characterizations and different estimation methods. Pakistan Journal of Statistics and Operation Research, 17(2), 399-425.
  53. Yousof, H. M., Ali, M. M., Goual, H. and Ibrahim. M. (2021b). A new reciprocal Rayleigh extension: properties, copulas, different methods of estimation and modified right censored test for validation, Statistics in Transition New Series, 23(3), 1-23.
  54. Yousof, H. M., Ali, M. M., Hamedani, G. G., Aidi, K. and Ibrahim, M. (2021c). A new lifetime distribution with properties, characterizations, validation testing, different estimation methods. Statistics, Optimization & Information Computing, forthcoming.
  55. Yousof, H. M., Altun, E., Rasekhi, M., Alizadeh, M., Hamedani, G. G. and Ali, M. M. (2019). A new lifetime model with regression models, characterizations and applications. Communications in Statistics-Simulation and Computation, 48(1), 264-286.
  56. Yousof, H. M., Majumder, M., Jahanshahi, S. M. A., Ali, M. M. and Hamedani G. G. (2018). A new Weibull class of distributions: theory, characterizations and applications, Journal of Statistical Research of Iran, 15, 45-83.
  57. Zheng, G. (2002). On the fisher information matrix in type II censored data from the exponentiated exponential family. Biom. J. 44, 353-357.
  58. Zheng, G. and Park, S. (2004). A note on time savings in censored life testing. J. Stat. Plan. Inference 124, 289-300.

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