New Extension of Weibull Distribution: Copula, Mathematical Properties and Data Modeling
Abstract
This paper introduces a new flexible four-parameter lifetime model. Various of its structural properties are derived. The new density is expressed as a linear mixture of well-known exponentiated Weibull density. The maximum likelihood method is used to estimate the model parameters. Graphical simulation results to assess the performance of the maximum likelihood estimation are performed. We proved empirically the importance and flexibility of the new model in modeling four various types of data.References
Al-babtain, A. A., Elbatal, I. snf Yousof, H. M. (2020a). A new three parameter Fr´echet model with mathematical properties and applications. Journal of Taibah University for Science, 14(1), 265-278.
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.
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.
Alizadeh, M., Jamal, F., Yousof, H. M., Khanahmadi, M. and Hamedani, G. G. (2020). 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.
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.
Alshkaki, R. (2020). A generalized modification of the Kumaraswamy distribution for modeling and analyzing real-life data. Statistics, Optimization & Information Computing, 8(2), 521-548.
Aryal, G. R., Ortega, E. M., Hamedani, G. G. and Yousof, H. M. (2017a). The Topp-Leone generated Weibull distribution: regression model, characterizations and applications, International Journal of Statistics and Probability, 6, 126-141.
Aryal, G. R. and Yousof, H. M. (2017b). The exponentiated generalized-G Poisson family of distributions. Economic Quality Control, 32(1), 1-17.
Bjerkedal, T. (1960). Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. American Journal of Hygiene, 72, 130–148.
Bourguignon, M., Silva, R.B. and Cordeiro, G.M. (2014). The Weibull-G family of probability distributions, Journal of Data Science 12, 53–68.
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.
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.
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.
Cordeiro, G. M., Yousof, H. M., Ramires, T. G. and Ortega, E. M. M. (2017). The Burr XII system of densities: properties, regression model and applications. Journal of Statistical Computation and Simulation, 88(3), 432-456.
Elbatal, I. and Aryal, G. (2013). On the transmuted additive Weibull distribution. Austrian Journal of Statistics, 42(2), 117-132.
Esmaeili, H., Lak, F., & Altun, E. (2020). The Ristic-Balakrishnan odd log-logistic family of distributions: Properties and
Applications. Statistics, Optimization & Information Computing, 8(1), 17-35.
Farlie, D. J. G. (1960) The performance of some correlation coefficients for a general bivariate distribution. Biometrika, 47, 307-323.
Ghitany, M. E., Al-Hussaini, E. K. and Al-Jarallah, R. A. (2005). Marshall–Olkin extended Weibull distribution and its application to censored data. Journal of Applied Statistics, 32(10), 1025-1034.
Gumbel, E. J. (1961). Bivariate logistic distributions. Journal of the American Statistical Association, 56(294), 335-349.
Gumbel, E. J. (1960) Bivariate exponential distributions. Journ. Amer. Statist. Assoc., 55, 698-707.
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. Pak. J. Stat. Oper. Res., 14(3), 737-758.
Hamedani G. G. Rasekhi, M., Najibi, S. M., Yousof, H. M. and Alizadeh, M., (2019). Type II general exponential class of distributions. Pak. J. Stat. Oper. Res., XV (2), 503-523.
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.
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 & Information Computing, 8(2), 610-630.
Ibrahim, M. and Yousof, H. M. (2020). A new generalized Lomax model: statistical properties and applications. Journal of Data Science, 18(1), 190-217.
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.
Ibrahim, M., Mohammed, W. and Yousof, H. M. (2020). Bayesian and Classical Estimation for the One Parameter Double Lindley Model. Pakistan Journal of Statistics and Operation Research, 409-420.
Johnson, N. L. and Kotz, S. (1975) On some generalized Farlie-Gumbel-Morgenstern distributions. Commun. Stat. Theory, 4, 415-427.
Johnson, N. L. and Kotz, S. (1977) On some generalized Farlie-Gumbel-Morgenstern distributions- II: Regression, correlation and further generalizations. Commun. Stat.Theory, 6, 485-496.
Khalil, M. G., Hamedani, G. G., & Yousof, H. M. (2019). The Burr X exponentiated Weibull model: Characterizations, mathematical properties and applications to failure and survival times data. Pakistan Journal of Statistics and Operation Research, 141-160.
Khan, M. N. (2015). The modified beta Weibull distribution. Hacettepe Journal of Mathematics and Statistics, 44, 1553-1568.
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.
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. Pakistan Journal of Statistics and Operation Research, 541-556.
Korkmaz, M. C., Yousof, H. M., Alizadeh, M. and Hamedani, G. G. (2019). The Topp-Leone generalized odd log-logistic family of distributions: properties, characterizations and applications. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 1506-1527.
Korkmaz, M. C., 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.
Korkmaz, M. C., Yousof, H. M., Hamedani, G. G. and Ali, M. M. (2018b). The Marshall-Olkin generalized G Poisson family of distributions. Pak. J. Statist, 34(3), 251-267.
Korkmaz, M. C., Yousof, H. M., & Hamedani, G. G. (2018c). The exponential Lindley odd log-logistic-G family: Properties, characterizations and applications. Journal of Statistical Theory and Applications, 17(3), 554-571.
Lee, E. T. and Wang, J. (2003). Statistical methods for survival data analysis (Vol. 476). John Wiley & Sons.
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.
Lehmann, E. L. (1953). The power of rank tests. Annals of Mathematical Statistics 24, 23-43.
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, 343-355.
Mansour, M. M., Butt, N. S., Ansari, S. I., Yousof, H. M., Ali, M. M., Ibrahim, M. (2020a). A new exponentiated Weibull
distribution’s extension: copula, mathematical properties and applications. Contributions to Mathematics, 1 (2020b) 57–66. DOI: 10.47443/cm.2020.0018
Mansour, M., Korkmaz, M. C., Ali, M. M., Yousof, H., 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.
Mansour, M., Rasekhi, M., Ibrahim, M., Aidi, K., Yousof, H. M. and Elrazik, E. A. (2020). A new parametric life distribution with modified Bagdonaviˇcius–Nikulin goodness-of-fit test for censored validation, properties, applications, and different estimation methods. Entropy, 22(5), 592.
Morgenstern, D. (1956). Einfache beispiele zweidimensionaler verteilungen. Mitteilingsblatt fur Mathematische Statistik, 8, 234-235.
Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42, 299-302.
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.
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.
Provost, S.B. Saboor, A. and Ahmad, M. (2011). The gamma-Weibull distribution, Pak. Journal Stat., 27, 111–131.
Ristic, M.M. and Balakrishnan, N. (2012). The gamma-exponentiated exponential distribution. Journal of Statistical Computation and Simulation, 82, 1191-1206.
Smith, R. L. and Naylor, J. (1987). A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution. Journal of the Royal Statistical Society: Series C (Applied Statistics), 36(3), 358-369.
Weibull, W. (1951). A statistical distribution function of wide applicability. J. Appl. Mech.-Trans, 18(3), 293-297.
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.
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.
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.
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.
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
Yousof, H. M., Korkmaz, M. C. and Sen, S. (2019). A new two-parameter lifetime model. Annals of Data Science, 1-16.
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.
Yousof, H. M., Rasekhi, M., Afify, A. Z., Alizadeh, M., Ghosh, I. and Hamedani G. G. (2017c). The beta Weibull-G family of distributions: theory, characterizations and applications. Pakistan Journal of Statistics, 33, 95-116.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).