Main Article Content

Abstract

After studying the mathematical properties of the Double Burr XII model, we present Bayesian and non-Bayesian estimation for its unknown parameters. Also, we constructed a new statistical test for goodness-of-fit in case of complete and censored samples. The modified test is developed based on the Nikulin-Rao-Robson statistic for validation. Simulations are performed for assessing the new test along with nine applications on real data.

Keywords

Barzilai-Borwein Algorithm; Bayesian Estimation; Validation Nikulin-Rao-Robson;

Article Details

How to Cite
Ibrahim, M., Ali, M. M., Goual, H., & Yousof, H. (2022). The Double Burr Type XII Model: Censored and Uncensored Validation Using a New Nikulin-Rao-Robson Goodness-of-Fit Test with Bayesian and Non-Bayesian Estimation Methods. Pakistan Journal of Statistics and Operation Research, 18(4), 901-927. https://doi.org/10.18187/pjsor.v18i4.3600

References

  1. Aboraya, M., Ali, M. M., Yousof, H. M. and Ibrahim, M. (2022). A novel Lomax extension with statistical properties, copulas, different estimation methods and applications. Bulletin of the Malaysian Mathematical Sciences Society, (2022), 1-36. DOI: https://doi.org/10.1007/s40840-022-01250-y
  2. Aboraya, M., M. Yousof, H. M., Hamedani, G. G. and Ibrahim, M. (2020). A new family of discrete distributions with mathematical properties, characterizations, Bayesian and non-Bayesian estimation methods. Mathematics, 8, 1648. DOI: https://doi.org/10.3390/math8101648
  3. Ahmed, B., Ali, M. M. and Yousof, H. M. (2022). A novel G family for single acceptance sampling plans with application in quality and risk decisions, Annals of Data Science, forthcoming. DOI: https://doi.org/10.1007/s40745-022-00451-3
  4. Ahmed, B. and Yousof, H. M. (2022a). A new group acceptance sampling plans based on percentiles for the Weibull Fréchet model. Statistics, Optimization & Information Computing, forthcoming.
  5. Ahmed, B., Chesneau, C., Ali, M. M. and Yousof, H. M. (2022b). Amputated Life Testing for Weibull-Fréchet Percentiles: Single, Double and Multiple Group Sampling Inspection Plans with Applications. Pakistan Journal of Statistics and Operation Research, forthcoming.
  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, 615-623. DOI: https://doi.org/10.18187/pjsor.v17i3.3888
  7. Al-babtain, A. A., Elbatal, I. and Yousof, H. M. (2020a). A new flexible three-parameter model: properties, Clayton copula, and modeling real data. Symmetry, 12, 440. DOI: https://doi.org/10.3390/sym12030440
  8. 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, 265-278.‏ DOI: https://doi.org/10.1080/16583655.2020.1733767
  9. Ali, M. M., But, N. S., Hamedani, G. G., Nadarajah, S., Yousof, H. M., and Ibrahim, M. (2022a). A New Compound G Family of Distributions: Properties, Copulas, Characterizations, Real Data Applications with Different Methods of Estimation, CRC Press, Taylor & Francis Group.
  10. Ali, M. M., Ibrahim, M. and Yousof, H. M. (2021a). Expanding the Burr X model: properties, copula, real data modeling and different methods of estimation. Optimal Decision Making in Operations Research & Statistics: Methodologies and Applications, CRC Press, 21-42.
  11. Ali, M. M., Ali, I., Yousof, H. M. and Ibrahim, M. (2022b). G Families of Probability Distributions: Theory and Practices. CRC Press, Taylor & Francis Group.
  12. Ali, M. M., Yousof, H. M. and Ibrahim, M. (2021b). A new version of the generalized Rayleigh distribution with copula, properties, applications and different methods of estimation. Optimal Decision Making in Operations Research & Statistics: Methodologies and Applications, CRC Press, 1-20.
  13. Alizadeh, M., Rasekhi, M., Yousof, H. M., Hamedani, G. and Ataei, A. (2022). The Odd Log-Logistic Transmuted-G Family of Distributions: Properties, Characterization, Applications and Different Methods of Estimation. Statistics, Optimization & Information Computing, 10(3), 904-924. DOI: https://doi.org/10.19139/soic-2310-5070-1182
  14. 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, 145-150.‏
  15. Alizadeh, M., Rasekhi, M., Yousof, H. M., Ramires, T. G. and Hamedani G. G. (2018). Extended exponentiated Nadarajah-Haghighi model: mathematical properties, characterizations and applications. Studia Scientiarum Mathematicarum Hungarica, 55, 498-522. DOI: https://doi.org/10.1556/012.2018.55.4.1408
  16. 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, 23, 1-27.‏ DOI: https://doi.org/10.1080/09720510.2019.1685228
  17. Almazah, M.M.A., Almuqrin, M.A., Eliwa, M.S., El-Morshedy, M., Yousof, H.M. (2021). Modeling Extreme Values Utilizing an Asymmetric Probability Function. Symmetry, 13, 1730. DOI: https://doi.org/10.3390/sym13091730
  18. Altun, E., Alizadeh, M., Kadilar, G. O. and Yousof, H. M. (2022) New Odd Log-Logistic Family of Distributions: Properties, Regression Models and Applications, CRC Press, Taylor & Francis Group.
  19. Altun, E., Yousof, H. M. and Hamedani, G. G. (2018a). A new log-location regression model with influence diagnostics and residual analysis. Facta Universitatis, Series: Mathematics and Informatics, 33, 417-449. DOI: https://doi.org/10.22190/FUMI1803417A
  20. Altun, E., Yousof, H. M., Chakraborty, S. and Handique, L. (2018b). Zografos-Balakrishnan Burr XII distribution: regression modeling and applications. International Journal of Mathematics and Statistics, 19, 46-70. DOI: https://doi.org/10.15672/HJMS.2017.410
  21. Aryal, G. R. and Yousof, H. M. (2017). The exponentiated generalized-G Poisson family of distributions. Economic Quality Control, 32, 1-17. DOI: https://doi.org/10.1515/eqc-2017-0004
  22. 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. DOI: https://doi.org/10.5539/ijsp.v6n1p126
  23. Burr, I. W. (1942). Cumulative frequency functions. Annals of Mathematical Statistics, 13, 215-232. DOI: https://doi.org/10.1214/aoms/1177731607
  24. Burr, I. W. (1968). On a general system of distributions, III. The simple range. Journal of the American Statistical Association, 63, 636-643. DOI: https://doi.org/10.1080/01621459.1968.11009282
  25. Burr, I. W. (1973). Parameters for a general system of distributions to match a grid of α_3 and α_4. Communications in Statistics, 2, 1-21. DOI: https://doi.org/10.1080/03610917308548217
  26. Burr, I. W. and Cislak, P. J. (1968). On a general system of distributions: I. Its curve-shaped characteristics; II. The sample median. Journalof the American Statistical Association, 63, 627-635. DOI: https://doi.org/10.1080/01621459.1968.11009281
  27. Chesneau, C. and Yousof, H. M. (2021). On a special generalized mixture class of probabilistic models. Journal of Nonlinear Modeling and Analysis, 3, 71-92.
  28. Chesneau, C. and Yousof, H. M. (2022). On the use of copulas to construct univariate generalized families of continuous distributions. G families of Probability Distributions Theory and Practices, CRC Press, Taylor & Francis Group.
  29. Chesneau, C., Yousof, H. M., Hamedani, G. and Ibrahim, M. (2022). A New One-parameter Discrete Distribution: The Discrete Inverse Burr Distribution: Characterizations, Properties, Applications, Bayesian and Non-Bayesian Estimations. Statistics, Optimization & Information Computing, 10, 352-371. DOI: https://doi.org/10.19139/soic-2310-5070-1393
  30. Elgohari, H. and Yousof, H. M. (2020a). A generalization of Lomax distribution with properties, copula and real data applications. Pakistan Journal of Statistics and Operation Research, 16, 697-711. DOI: https://doi.org/10.18187/pjsor.v16i4.3260
  31. Elgohari, H. and Yousof, H. M. (2021b). A new extreme value model with different copula, statistical properties and applications. Pakistan Journal of Statistics and Operation Research, 17, 1015-1035. DOI: https://doi.org/10.18187/pjsor.v17i4.3471
  32. Elgohari, H. and Yousof, H. M. (2020c). New extension of Weibull distribution: copula, mathematical properties and data modeling. Statistics, Optimization & Information Computing, 8, 972-993. DOI: https://doi.org/10.19139/soic-2310-5070-1036
  33. Elgohari, H., Ibrahim, M. and Yousof, H. M. (2021). A new probability distribution for modeling failure and service times: properties, copulas and various estimation methods. Statistics, Optimization & Information Computing, 8(3), 555-586. DOI: https://doi.org/10.19139/soic-2310-5070-1101
  34. Elsayed, H. A. and Yousof, H. M. (2021). Extended Poisson generalized Burr XII distribution. Journal of Applied Probability and Statistics, 16(1), 01-30.‏
  35. 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, 745-771. DOI: https://doi.org/10.18187/pjsor.v15i3.2675
  36. 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, 393-423. DOI: https://doi.org/10.1080/02664763.2019.1639642
  37. 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, 13(6), 330-353. DOI: https://doi.org/10.22436/jnsa.013.06.04
  38. Hamedani, G. G., Altun, E, Korkmaz, M. C., Yousof, H. M. and Butt, N. S. (2018). A new extended G family of continuous distributions with mathematical properties, characterizations and regression modeling. Pakistan Journal of Statistics and Operation Research, 14, 737-758. DOI: https://doi.org/10.18187/pjsor.v14i3.2484
  39. Hamedani, G. G., Korkmaz, M. Ç., Butt, N. S. and Yousof H. M. (2022). The Type II Quasi Lambert G Family of Probability Distributions. Pakistan Journal of Statistics and Operation Research, forthcoming. DOI: https://doi.org/10.18187/pjsor.v17i3.3562
  40. Hamedani, G. G. Rasekhi, M., Najib, S. M., Yousof, H. M. and Alizadeh, M., (2019). Type II general exponential class of distributions. Pakistan Journal of Statistics and Operation Research, 15, 503-523. DOI: https://doi.org/10.18187/pjsor.v15i2.1699
  41. Hamedani, G. G. Yousof, H. M., Rasekhi, M., Alizadeh, M., Najibi, S. M. (2017). Type I general exponential class of distributions. Pakistan Journal of Statistics and Operation Research, 14, 39-55. DOI: https://doi.org/10.18187/pjsor.v14i1.2193
  42. Ibrahim, M., Aidi, K., Ali, M. M. and Yousof, H. M. (2022a). A novel test statistic for right censored validity under a new Chen extension with applications in reliability and medicine. Annals of Data Science, forthcoming. DOI: https://doi.org/10.1007/s40745-022-00416-6
  43. Ibrahim, M., Ali, M. M. and Yousof, H. M. (2022b). The discrete analogue of the Weibull G family: properties, different applications, Bayesian and non-Bayesian estimation methods. Annals of Data Science, forthcoming. DOI: https://doi.org/10.1007/s40745-021-00327-y
  44. Ibrahim, M., Altun, E. and Yousof, H. M. (2020a). A new distribution for modeling lifetime data with different methods of estimation and censored regression modeling. Statistics, Optimization & Information Computing, 8(2), 610-630.‏ DOI: https://doi.org/10.19139/soic-2310-5070-678
  45. Ibrahim, M., Altun, E., Goual, H., and Yousof, H. M. (2020b). Modified goodness-of-fit type test for censored validation under a new Burr type XII distribution with different methods of estimation and regression modeling. Eurasian Bulletin of Mathematics, 3(3), 162-182.‏
  46. Ibrahim. M., Handique, L., Chakraborty, S., Butt, N. S. and M. Yousof, H. (2021). A new three-parameter xgamma Fréchet distribution with different methods of estimation and applications. Pakistan Journal of Statistics and Operation Research, 17(1), 291-308. DOI: https://doi.org/10.18187/pjsor.v17i1.2887
  47. Ibrahim, M. and Yousof, H. M. (2020). Transmuted Topp-Leone Weibull lifetime distribution: Statistical properties and different method of estimation. Pakistan Journal of Statistics and Operation Research, 501-515. DOI: https://doi.org/10.18187/pjsor.v16i3.2811
  48. 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, 473-495. DOI: https://doi.org/10.29220/CSAM.2019.26.5.473
  49. 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, 121-161.‏ DOI: https://doi.org/10.29252/jirss.19.1.121
  50. 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, 59-74.‏ DOI: https://doi.org/10.2991/jsta.d.200302.001
  51. Korkmaz, M. C. Yousof, H. M. and Hamedani G. G. (2018a). The exponential Lindley odd log-logistic G family: properties, characterizations and applications. Journal of Statistical Theory and Applications, 17, 554 - 571. DOI: https://doi.org/10.2991/jsta.2018.17.3.10
  52. Korkmaz, M. C., Yousof, H. M., Hamedani G. G. and Ali, M. M. (2018b). The Marshall–Olkin generalized G Poisson family of distributions, Pakistan Journal of Statistics, 34, 251-267.
  53. 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, 1508. DOI: https://doi.org/10.3390/math8091508
  54. 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, 57–66. DOI: https://doi.org/10.47443/cm.2020.0018
  55. 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, 84-102.
  56. 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, 592. DOI: https://doi.org/10.3390/e22050592
  57. 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, 223-238. DOI: https://doi.org/10.22436/jnsa.013.05.01
  58. 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, 373-386. DOI: https://doi.org/10.18187/pjsor.v16i2.3298
  59. Merovci, F., Yousof, H. M. 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, 343-355.‏ DOI: https://doi.org/10.18187/pjsor.v16i2.3230
  60. 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, 10800-10822. DOI: https://doi.org/10.1080/03610926.2016.1248782
  61. Mohamed, H. S., Cordeiro, G. M., and Yousof, H. M. (2022a). The synthetic autoregressive model for the insurance claims payment data: modeling and future prediction. Optimization & Information Computing, forthcoming.
  62. Mohamed, H. S., Cordeiro, G. M., Minkah, R., Yousof, H. M. and Ibrahim, M. (2022b). A size-of-loss model for the negatively skewed insurance claims data: applications, risk analysis using different methods and statistical forecasting. Journal of Applied Statistics, forthcoming. DOI: https://doi.org/10.1080/02664763.2022.2125936
  63. Mohamed, H. S., Ali, M. M. and Yousof, H. M. (2022c). The Lindley Gompertz Model for Estimating the Survival Rates: Properties and Applications in Insurance, Annals of Data Science, forthcoming. DOI: https://doi.org/10.1007/s40745-022-00450-4
  64. Nascimento, A. D. C., Silva, K. F., Cordeiro, G. M., Alizadeh, M. and Yousof, H. M. (2019). The odd Nadarajah-Haghighi family of distributions: properties and applications. Studia Scientiarum Mathematicarum Hungarica, 56, 1-26. DOI: https://doi.org/10.1556/012.2019.56.2.1416
  65. Nasir, M. A., Korkmaz, M. C., Jamal, F. and Yousof, H. M. (2018). On a new Weibull Burr XII distribution for lifetime data. Sohag Journal of Mathematics, 5(2), 47-56.‏ DOI: https://doi.org/10.18576/sjm/050202
  66. Nikulin. M.S, (1973a). Chi-squared test for normality. in proceedings of the International Vilnius Conference on Probability Theory and Mathematical Statistics, 2, 119-122.
  67. Nikulin. M.S, (1973b). Chi-squared test for continuous distributions with shift and scale parameters, Theory of Probability and its Aplications. 18, 559-568. DOI: https://doi.org/10.1137/1118069
  68. Nikulin. M.S, (1973c). On a Chi-squared test for continuous distributions, Theory of Probability and its Aplications. 19, 638-639.
  69. Paranaíba, P. F., Ortega, E. M., Cordeiro, G. M., & Pescim, R. R. (2011). The beta Burr XII distribution with application to lifetime data. Computational statistics & data analysis, 55(2), 1118-1136.‏ DOI: https://doi.org/10.1016/j.csda.2010.09.009
  70. Paranaíba, P. F., Ortega, E. M., Cordeiro, G. M., & Pascoa, M. A. D. (2013). The Kumaraswamy Burr XII distribution: theory and practice. Journal of Statistical Computation and Simulation, 83(11), 2117-2143.‏ DOI: https://doi.org/10.1080/00949655.2012.683003
  71. Rao, K. C., Robson, D. S. (1974). A Chi-square statistic for goodness-of-fit tests within the exponential family. Communication in Statistics, 3, 1139-1153. DOI: https://doi.org/10.1080/03610917408548327
  72. Rodriguez, R.N. (1977). A guide to the Burr type XII distributions. Biometrika, 64, 129-134. DOI: https://doi.org/10.1093/biomet/64.1.129
  73. Saber, M. M., Hamedani, G. G., Yousof, H. M. But, N. S., Ahmed, B. and Yousof, H. M. (2022a) A Family of Continuous Probability Distributions: Theory, Characterizations, Properties and Different Copulas, CRC Press, Taylor & Francis Group.
  74. Saber, M. M. and Yousof, H. M. (2022). Bayesian and classical inference for generalized stress-strength parameter under generalized logistic distribution, Statistics, Optimization & Information Computing, forthcoming.
  75. Saber, M. M. Marwa M. Mohie El-Din and Yousof, H. M. (2022b). Reliability estimation for the remained stress-strength model under the generalized exponential lifetime distribution, Journal of Probability and Statistics, 2021, 1-10. DOI: https://doi.org/10.1155/2021/7363449
  76. Salah, M. M., El-Morshedy, M., Eliwa, M. S. and Yousof, H. M. (2020). Expanded Fréchet Model: Mathematical Properties, Copula, Different Estimation Methods, Applications and Validation Testing. Mathematics, 8(11), 1949. DOI: https://doi.org/10.3390/math8111949
  77. Salem M., Butt. N. S., and Yousof, H. M. (2022). Short-Term Insurance Claims Payments Forecasting with Holt-Winter Filtering and Residual Analysis. Pakistan Journal of Statistics and Operation Research, forthcoming.
  78. Shehata, W. A. M. and 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, 649-667. DOI: https://doi.org/10.18187/pjsor.v17i3.3311
  79. Shehata, W. A. M. and Yousof, H. M. (2022). A novel two-parameter Nadarajah-Haghighi extension: properties, copulas, modeling real data and different estimation methods. Statistics, Optimization & Information Computing, 10, 725-749. DOI: https://doi.org/10.19139/soic-2310-5070-1250
  80. Shehata, W. A. M., Butt, N. S., Yousof, H. and 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, 249-272. DOI: https://doi.org/10.18187/pjsor.v18i1.3930
  81. Shehata, W. A. M., Yousof, H. M. and Aboraya, M. (2021). A novel generator of continuous probability distributions for the asymmetric left-skewed bimodal real-life data with properties and copulas. Pakistan Journal of Statistics and Operation Research, 17, 943-961. DOI: https://doi.org/10.18187/pjsor.v17i4.3903
  82. Tadikamalla, P. R. (1980). A look at the Burr and related distributions, International Statistical Review, 48, 337-344. DOI: https://doi.org/10.2307/1402945
  83. 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, 57. DOI: https://doi.org/10.3390/sym12010057
  84. Yadav, A. S., Shukla, S., Goual, H., Saha, M. and Yousof, H. M. (2022). Validation of xgamma exponential model via Nikulin-Rao-Robson goodness-of- fit test under complete and censored sample with different methods of estimation. Statistics, Optimization & Information Computing, 10, 457-483. DOI: https://doi.org/10.19139/soic-2310-5070-1107
  85. Yousof, H. M., Afify, A. Z., Abd El Hadi, N. E., Hamedani, G. G. and Butt, N. S. (2016). On six-parameter Fréchet distribution: properties and applications. Pakistan Journal of Statistics and Operation Research, 12, 281-299.‏ DOI: https://doi.org/10.18187/pjsor.v12i2.1327
  86. Yousof, H. M., Afify, A. Z., Nadarajah, S., Hamedani, G. and Aryal, G. R. (2018a). The Marshall-Olkin generalized-G family of distributions with Applications. Statistica, 78, 273-295.‏
  87. 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. DOI: https://doi.org/10.18187/pjsor.v17i2.3631
  88. Yousof, H. M., Ali, M. M., Hamedani, G. G., Aidi, K. and Ibrahim, M. (2022). A new lifetime distribution with properties, characterizations, validation testing, different estimation methods. Statistics, Optimization & Information Computing, 10, 519-547. DOI: https://doi.org/10.19139/soic-2310-5070-1222
  89. Yousof, H. M., Ahsanullah, M., & Khalil, M. G. (2019a). A new zero-truncated version of the Poisson Burr XII distribution: characterizations and properties. Journal of Statistical Theory and Applications, 18(1), 1-11. DOI: https://doi.org/10.2991/jsta.d.190306.001
  90. Yousof, H. M., Ali, M. M., Cordeiro, G. M., Hamedani, G. G. and Ibrahim, M. (2022). A Novel Family of Continuous Distributions: Properties, Characterizations, Statistical Modeling and Different Estimation Methods, CRC Press, Taylor & Francis Group.
  91. 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. DOI: https://doi.org/10.21307/stattrans-2021-029
  92. Yousof, H. M., Alizadeh, M., Jahanshahiand, S. M. A., Ramires, T. G., Ghosh, I. and Hamedani, G. G. (2017). 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
  93. Yousof, H. M., Altun, E., Ramires, T. G., Alizadeh, M. and Rasekhi, M. (2018b). A new family of distributions with properties, regression models and applications, Journal of Statistics and Management Systems, 21, 163-188. DOI: https://doi.org/10.1080/09720510.2017.1411028
  94. Yousof, H. M., Altun, E., Rasekhi, M., Alizadeh, M., Hamedani, G. G. and Ali, M. M. (2019b). A new lifetime model with regression models, characterizations and applications. Communications in Statistics-Simulation and Computation, 48, 264-286. DOI: https://doi.org/10.1080/03610918.2017.1377241
  95. Yousof, H. M., Chesneau, C., Hamedani, G. and Ibrahim, M. (2021c). A new discrete distribution: properties, characterizations, modeling real count data, Bayesian and non-Bayesian estimations. Statistica, 81, 135-162.
  96. Yousof, H. M., Hamedani, G. G. and Ibrahim, M. (2020). The Two-parameter Xgamma Fréchet Distribution: Characterizations, Copulas, Mathematical Properties and Different Classical Estimation Methods. Contributions to Mathematics, 2 (2020), 32-41. DOI: https://doi.org/10.47443/cm.2020.0031