Main Article Content

Abstract

Probability distributions and their families play an effective role in statistical modeling and statistical analysis. Recently, researchers have been increasingly interested in generating new families with high flexibility and low number of milestones. We propose and study a new family of continuous distributions. Relevant properties are presented. Many bivariate versions of the new family are derived under the Farlie-Gumbel-Morgenstern copula, modified Farlie-Gumbel-Morgenstern copula, Clayton copula, entropy copula and Ali-Mikhail-Haq copula. We present two characterizations of the new family. Different estimation methods such as the maximum likelihood estimation, maximum product spacing estimation, least squares estimation, weighted least squares estimation, Anderson-Darling estimation and the Cramer-von Mises estimation methods are considered. Simulation studies for comparing estimation methods are performed based on the baseline Lindley model. Two real data sets are analyzed for comparing the competitive models.

Keywords

Characterizations Maximum Product Spacing Maximum Likelihood Clayton copula Anderson-Darling Estimation Ali-Mikhail-Haq Copula

Article Details

How to Cite
Hamedani, G., Korkmaz, M. C., Butt, N. S., & Yousof, H. M. (2022). The Type II Quasi Lambert Family. Pakistan Journal of Statistics and Operation Research, 18(4), 963-983. https://doi.org/10.18187/pjsor.v18i4.3907

References

  1. Ahmed, B. and Yousof, H. M. (2022). A New Group Acceptance Sampling Plans based on Percentiles for the Weibull Fréchet Model. Statistics, Optimization & Information Computing, forthcoming.
  2. Ahmed, B., Ali, M. M. and Yousof, H. M. (2022a). A Novel G Family for Single Acceptance Sampling Plan with Application in Quality and Risk Decisions, Annals of Data Science, forthcoming. DOI: https://doi.org/10.1007/s40745-022-00451-3
  3. 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.
  4. Ali, M. M., Mikhail, N. N. and Haq, M. S. (1978). A class of bivariate distributions including the bivariate logistic. Journal of multivariate analysis, 8(3), 405-412. DOI: https://doi.org/10.1016/0047-259X(78)90063-5
  5. 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. Journal of Data Science, 15(3), 443-465. DOI: https://doi.org/10.6339/JDS.201707_15(3).0005
  6. 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, 1-20. DOI: https://doi.org/10.15672/HJMS.2017.440
  7. Alzaatreh A, Lee C and Famoye F (2013). A New Method for Generating Families of Continuous Distributions. Metron, 71(1), 63-79. DOI: https://doi.org/10.1007/s40300-013-0007-y
  8. Aryal, G. R., and Yousof, H. M. (2017). The exponentiated generalized-G Poisson family of distributions. Stochastics and Quality Control, 32(1), 7-23. DOI: https://doi.org/10.1515/eqc-2017-0004
  9. Bebbington, M., Lai, C. D. and Zitikis, R. (2007). A flexible Weibull extension. Reliability Engineering & System Safety, 92(6), 719-726. DOI: https://doi.org/10.1016/j.ress.2006.03.004
  10. Brito, E., Cordeiro, G. M., Yousof, H. M., Alizadeh, M. & 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
  11. Cakmakyapan, S. and Kadilar, G. O. (2014). A new customer lifetime duration distribution: the Kumaraswamy Lindley distribution. International Journal of Trade, Economics and Finance, 5(5), 441. DOI: https://doi.org/10.7763/IJTEF.2014.V5.412
  12. Cakmakyapan, S. and Ozel. G. (2017). The Lindley family of distributions: Properties and applications. Hacettepe Journal of Mathematics and Statistics, 46(6), 1113-1137. DOI: https://doi.org/10.15672/HJMS.201611615850
  13. Cheng, R. C. H. and Amin, N. A. K. (1979). Maximum product of spacings estimation with application to the lognormal distribution. Math Report, 791.
  14. 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(8), 1399-1429. DOI: https://doi.org/10.1016/j.jfranklin.2010.06.010
  15. 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
  16. 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(4), 697-711. DOI: https://doi.org/10.18187/pjsor.v16i4.3260
  17. Elgohari, H. and Yousof, H. M. (2020b). New Extension of Weibull Distribution: Copula, Mathematical Properties and Data Modeling. Statistics, Optimization & Information Computing, 8(4), 972-993. DOI: https://doi.org/10.19139/soic-2310-5070-1036
  18. Elgohari, H. and Yousof, H. M. (2021). A New Extreme Value Model with Different Copula, Statistical Properties and Applications. Pakistan Journal of Statistics and Operation Research, 17(4), 1015-1035. DOI: https://doi.org/10.18187/pjsor.v17i4.3471
  19. 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
  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.‏ DOI: https://doi.org/10.18187/pjsor.v15i3.2675
  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.‏ DOI: https://doi.org/10.1080/02664763.2019.1639642
  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. DOI: https://doi.org/10.22436/jnsa.013.06.04
  23. Famoye, F., Lee, C. and Olumolade, O. (2005). The beta-Weibull distribution. Journal of Statistical Theory and Applications, 4(2), 121-136.
  24. Farlie, D. J. G. (1960) The performance of some correlation coefficients for a general bivariate distribution. Biometrika, 47, 307-323. DOI: https://doi.org/10.1093/biomet/47.3-4.307
  25. Feigl, P., and Zellen, M. (1965). Estimation of Exponential Survival Probabilities with Concomitant Information, Biometrics, Vol. 21, No. 4, pp. 826-838 DOI: https://doi.org/10.2307/2528247
  26. Galambos, J. and Kotz, S. Characterizations of probability distributions. A unified approach with emphasis on exponential and related models, Lecture Notes in Mathematics, p.675. Springer, Berlin (1978).
  27. Glannzel, W, Telcs, A, Schubert, A. Characterization by truncated moments and its application to Pearson-type distributions, Z. Wahrsch. Verw. Gebiete 66, 173-182 (1984). DOI: https://doi.org/10.1007/BF00531527
  28. Glanzel, W. and Hamedani, G.G. Characterizations of the univariate distributions, Studia Scien. Math. hung., 37, 83-118 (2001). DOI: https://doi.org/10.1556/sscmath.37.2001.1-2.5
  29. Glanzel, W., A characterization theorem based on truncated moments and its application to some distribution families, Mathematical Statistics and Probability Theory (Bad Tatzmannsdorf, 1986), Vol. B, Reidel, Dordrecht, (1987), 75–84. DOI: https://doi.org/10.1007/978-94-009-3965-3_8
  30. Glanzel, W., Some consequences of a characterization theorem based on truncated moments, Statistics: A Journal of Theoretical and Applied Statistics, 21(4), (1990), 613–618. DOI: https://doi.org/10.1080/02331889008802273
  31. Gleaton, J. U. and Lynch, J. D. (2006). Properties of generalized loglogistic families of lifetime distributions. Journal of Probability and Statistical Science, 4(1), 51-64.
  32. Gupta, R. C., Gupta, P. L. and Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives. Communications in Statistics-Theory and methods, 27(4), 887-904. DOI: https://doi.org/10.1080/03610929808832134
  33. Hamed, M. S., Cordeiro, G. M. and Yousof, H. M. (2022). A New Compound Lomax Model: Properties, Copulas, Modeling and Risk Analysis Utilizing the Negatively Skewed Insurance Claims Data. Pakistan Journal of Statistics and Operation Research, 18(3), 601-631. DOI: https://doi.org/10.18187/pjsor.v18i3.3652
  34. Hamedani G. G. Yousof, H. M., Rasekhi, M., Alizadeh, M. & Najibi, S. M. (2017). Type I general exponential class of distributions. Pak. J. Stat. Oper. Res., XIV(1), 39-55. DOI: https://doi.org/10.18187/pjsor.v14i1.2193
  35. Hamedani, G. G., Altun, E., Korkmaz, M. C., Yousof, H. M., & 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, 737-758. DOI: https://doi.org/10.18187/pjsor.v14i3.2484
  36. Hamedani, G. G., Korkmaz, M. C., Butt, N. S. and Yousof, H. M. (2021). The Type I Quasi Lambert Family: Properties, Characterizations and Different Estimation Methods. Pakistan Journal of Statistics and Operation Research, 17(3), 545-558. DOI: https://doi.org/10.18187/pjsor.v17i3.3562
  37. Hamedani, G. G., Rasekhi, M., Najibi, S., Yousof, H. M., & Alizadeh, M. (2019). Type II general exponential class of distributions. Pakistan Journal of Statistics and Operation Research, 503-523. DOI: https://doi.org/10.18187/pjsor.v15i2.1699
  38. Hamedani, G.G., On certain generalized gamma convolution distributions II, Technical Report, No. 484, MSCS, Marquette University, 2013.
  39. Ibrahim, M., Ali, M. M., Goual, H. and Yousof, H. M. (2022a). Censored and uncensored validation for the double Burr XII model using a new Nikulin-Rao-Robson goodness-of-fit test with Bayesian and non-Bayesian estimation, Pakistan Journal of Statistics and Operation Research, forthcoming.
  40. Ibrahim, M., Hamedani, G. G., Butt, N. S. and Yousof, H. M. (2022b). Expanding the Nadarajah Haghighi Model: Copula, Censored and Uncensored Validation, Characterizations and Applications. Pakistan Journal of Statistics and Operation Research, 18(3), 537-553. DOI: https://doi.org/10.18187/pjsor.v18i3.3420
  41. Karamikabir, H., Afshari, M., Yousof, H. M., Alizadeh, M. and Hamedani, G. (2020). The Weibull Topp-Leone generated family of gistributions: statistical properties and applications. Journal of The Iranian Statistical Society, 19(1), 121-161. DOI: https://doi.org/10.29252/jirss.19.1.121
  42. Kim, J.H. and Jeon, Y. Credibility theory based on trimming, Insur. Math. Econ. 53(1), 36-47 (2013). DOI: https://doi.org/10.1016/j.insmatheco.2013.03.012
  43. 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. DOI: https://doi.org/10.18187/pjsor.v14i3.2598
  44. Korkmaz, M. C., Altun, E., Yousof, H. M. and Hamedani G. G. (2019). The odd power Lindley generator of probability distributions: properties, characterizations and regression modeling, International Journal of Statistics and Probability, 8(2), 70-89. DOI: https://doi.org/10.5539/ijsp.v8n2p70
  45. Korkmaz, M. C., 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. DOI: https://doi.org/10.2991/jsta.2018.17.3.10
  46. 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.
  47. Kotz, S. and Shanbhag, D.N. Some new approach to probability distributions, Adv. Appl. Probab. 12, 903-921 (1980). DOI: https://doi.org/10.1017/S0001867800020164
  48. Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society. Series B (Methodological), 102-107. DOI: https://doi.org/10.1111/j.2517-6161.1958.tb00278.x
  49. Lucena, S. E., Silva, A. H. A. and Cordeiro, G. M. (2015). The transmuted generalized gamma distribution: Properties and application. Journal of Data Science, 13(1), 187-206. DOI: https://doi.org/10.6339/JDS.201501_13(1).0010
  50. 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.‏ DOI: https://doi.org/10.3390/math8091508
  51. 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. DOI: https://doi.org/10.47443/cm.2020.0018
  52. 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.‏
  53. Mansour, M., Rasekhi, M., Ibrahim, M., Aidi, K., Yousof, H. M. and Elrazik, E. A. (2020d). A New Parametric Life Distribution with Modified Bagdonavicius–Nikulin Goodness-of-Fit Test for Censored Validation, Properties, Applications, and Different Estimation Methods. Entropy, 22(5), 592.‏ DOI: https://doi.org/10.3390/e22050592
  54. 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. DOI: https://doi.org/10.22436/jnsa.013.05.01
  55. 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.‏ DOI: https://doi.org/10.18187/pjsor.v16i2.3298
  56. Mead, M. E. (2014). The beta exponentiated Burr XII distribution. Journal of Statistics: Advances in Theory and Applications, Vol. 12, No. 2, 53-73.
  57. Merovci, F. and Sharma, V. K. (2014). The beta-Lindley distribution: properties and applications. Journal of Applied Mathematics, 2014. DOI: https://doi.org/10.1155/2014/198951
  58. 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. DOI: https://doi.org/10.1080/03610926.2016.1248782
  59. Mohamed, H. S., Ali, M. M. and Yousof, H. M. (2022a). The Lindley Gompertz Model for Estimating the Survival Rates: Properties and Applications in Insurance, Annals of Data Science, 10.1007/s40745-022-00451-3 DOI: https://doi.org/10.1007/s40745-022-00450-4
  60. 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
  61. Mohamed, H. S., Cordeiro, G. M. and Yousof, H. M. (2022c). The synthetic autoregressive model for the insurance claims payment data: modeling and future prediction. Statistics, Optimization & Information Computing, forthcoming.
  62. Morgenstern, D. (1956). Einfache beispiele zweidimensionaler verteilungen. Mitteilingsblatt fur Mathematische Statistik, 8, 234-235.
  63. 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(2), 1-26. DOI: https://doi.org/10.1556/012.2019.56.2.1416
  64. Ozel, G., Alizadeh, M., Cakmakyapan, S., Hamedani, G. G., Ortega, E. M. and Cancho, V. G. (2017). The odd log-logistic Lindley Poisson model for lifetime data. Communications in Statistics-Simulation and Computation, 46(8), 6513-6537. DOI: https://doi.org/10.1080/03610918.2016.1206931
  65. 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. DOI: https://doi.org/10.1063/1.3573634
  66. Rasekhi, M., Saber, M. M. and Yousof, H. M. (2020). Bayesian and classical inference of reliability in multicomponent stress-strength under the generalized logistic model. Communications in Statistics-Theory and Methods, 50(21), 5114-5125. DOI: https://doi.org/10.1080/03610926.2020.1726958
  67. 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
  68. Rodriguez-Lallena, J. A. and Ubeda-Flores, M. (2004). A new class of bivariate copulas. Statistics and Probability Letters, 66, 315–25. Salman S., M. and Prayoto, S. (1999). Total time on test plot analysis for mechanical components of the RSG-GAS reactor. Atom Indones, 25(2), 155-61. DOI: https://doi.org/10.1016/j.spl.2003.09.010
  69. 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.
  70. Saber, M. M. Marwa M. Mohie El-Din and Yousof, H. M. (2022a). 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
  71. Saber, M. M., Rasekhi, M. and Yousof, H. M. (2022b). Generalized Stress-Strength and Generalized Multicomponent Stress-Strength Models, Statistics, Optimization & Information Computing, forthcoming.
  72. 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
  73. 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.
  74. 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(1), 249-272. DOI: https://doi.org/10.18187/pjsor.v18i1.3930
  75. Shehata, W. A. M. and Yousof, H. M. (2021a). The four-parameter exponentiated Weibull model with Copula, properties and real data modeling. Pakistan Journal of Statistics and Operation Research, 17(3), 649-667. DOI: https://doi.org/10.18187/pjsor.v17i3.3311
  76. Shehata, W. A. M. and Yousof, H. M. (2021b). A novel two-parameter Nadarajah-Haghighi extension: properties, copulas, modeling real data and different estimation methods. Statistics, Optimization & Information Computing, forthcoming. DOI: https://doi.org/10.19139/soic-2310-5070-1250
  77. Salman S., M. and Prayoto, S. (1999). Total time on test plot analysis for mechanical components of the RSG-GAS reactor. Atom Indones, 25(2), 155-61.
  78. Topp, C. W. and Leone, F. C. (1955). A family of J-shaped frequency functions. Journal of the American Statistical Association, 50(269), 209-219. DOI: https://doi.org/10.1080/01621459.1955.10501259
  79. Weibull, W. (1951). Wide applicability. Journal of applied mechanics, 103(730), 293-297. DOI: https://doi.org/10.1115/1.4010337
  80. 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
  81. Yousof, H. M., Afify, A. Z., Alizadeh, M., Hamedani G. G., Jahanshahi, S. M. A. and Ghosh, I. (2018a). The generalized transmuted Poisson-G family of Distributions. Pak. J. Stat. Oper. Res., 14 (4), 759-779. DOI: https://doi.org/10.18187/pjsor.v14i4.2527
  82. Yousof, H. M., Afify, A. Z., Alizadeh, M., Nadarajah, S., Aryal, G. R. and Hamedani, G. G. (2018b). The Marshall-Olkin generalized-G family of distributions with Applications, STATISTICA, 78(3), 273- 295.
  83. Yousof, H. M., Alizadeh, M., Jahanshahi and, S. M. A., Ramires, T. G., Ghosh, I. and Hamedani G. G. (2017a). 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
  84. Yousof, H. M., Altun, E., Ramires, T. G., Alizadeh, M. and Rasekhi, M. (2018c). A new family of distributions with properties, regression models and applications, Journal of Statistics and Management Systems, 21(1), 163-188. DOI: https://doi.org/10.1080/09720510.2017.1411028
  85. Yousof, H. M., Majumder, M., Jahanshahi, S. M. A., Ali, M. M. and Hamedani G. G. (2018d). 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
  86. Yousof, H. M., Rasekhi, M., Afify, A. Z., Alizadeh, M., Ghosh, I. and Hamedani G. G. (2017b). The beta Weibull-G family of distributions: theory, characterizations and applications, Pakistan Journal of Statistics, 33, 95-116.
  87. Yousof, H., Mansoor, M., Alizadeh, M., Afify, A. and Ghosh, I. (2020). The Weibull-G Poisson family for analyzing lifetime data. Pakistan Journal of Statistics and Operation Research, 131-148. DOI: https://doi.org/10.18187/pjsor.v16i1.2840

Most read articles by the same author(s)

> >>