# A New Probability Distribution for Modeling Failure and Service Times: Properties, Copulas and Various Estimation Methods

### Abstract

In this paper, a new generalization of the Pareto type II model is introduced and studied. The new density canbe “right skewed” with heavy tail shape and its corresponding failure rate can be “J-shape”, “decreasing” and “upside down (or increasing-constant-decreasing)”. The new model may be used as an “under-dispersed” and “over-dispersed” model. Bayesian and non-Bayesian estimation methods are considered. We assessed the performance of all methods via simulation study. Bayesian and non-Bayesian estimation methods are compared in modeling real data via two applications. In modeling real data, the maximum likelihood method is the best estimation method. So, we used it in comparing competitive models. Before using the the maximum likelihood method, we performed simulation experiments to assess the finite sample behavior of it using the biases and mean squared errors.### References

Abouelmagd, T. H. M., Hamed, M. S., Hamedani, G. G., Ali, M. M., Goual, H., Korkmaz, M. C. and Yousof, H. M. (2019a). The zero truncated Poisson Burr X family of distributions with properties, characterizations, applications, and validation test. Journal of Nonlinear Sciences and Applications, 12(5), 314-336.

Abouelmagd, T. H. M., Hamed, M. S., Handique, L., Goual, H., Ali, M. M., Yousof, H. M. and Korkmaz, M. C. (2019b). A new class of distributions based on the zero truncated Poisson distribution with properties and applications. Journal of Nonlinear Sciences & Applications (JNSA), 12(3), 152-164.

Al-babtain,A.A.,Elbatal,I.,&Yousof,H.M.(2020). A new flexible three-parameter model: properties, clayton copula, and modeling real data. Symmetry, 12(3), 440.

Ali, M. M., Yousof, H. M. and Ibrahim, M. (2021a). 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. To appear.

Ali, M. M., Ibrahim, M. and Yousof, H. M. (2021b). 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. To appear.

Alizadeh, M., Cordeiro, G. M., Nascimento, A. D., Lima, M. D. C. S. and Ortega, E. M. (2017a). Odd-Burr generalized family of distributions with some applications. Journal of statistical computation and simulation, 87(2), 367-389.

Alizadeh, M., Ghosh, I., Yousof, H. M., Rasekhi, M. and Hamedani, G. G. (2017b). The generalized odd generalized exponential family of distributions: properties, characterizations and applications. Journal of Data Science, 15(3), 443-465.

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., Ramires, T. G. and Hamedani, G. G. (2018). Extended exponentiated Nadarajah-Haghighi model: Mathematical properties, characterizations and applications. Studia Scientiarum Mathematicarum Hungarica, 55(4), 498- 522.

Atkinson, A. B. and Harrison, A. J. (1978). Distribution of Personal Wealth in Britain (Cambridge University Press, Cambridge).

Altun, E. Alizadeh, M., Yousof, H. M., Rasekhi, M. and Hamedani, G. G. (2021a). A New Type II Half Logistic-G family of Distributions with Properties, Regression Models, Reliability Systems and Applications. Applications and Applied Mathematics: An International Journal. forthcoming.

Altun, E., Yousof, H. M. and Hamedani G. G. (2018a). A flexible extension of generalized half-normal distribution: characterizations and regression models. International Journal of Applied Mathematics and Statistics, 57(3), 27-49.

Altun, E., Yousof, H. M. and Hamedani G. G. (2018b). A new flexible extension of the generalized half-normal lifetime model with characterizations and regression modeling. Bulletin of Computational Applied Mathematics, 6(1), 83-115.

Altun, E., Yousof, H. M. and Hamedani, G. G. (2018c). A new generalization of generalized half-normal distribution: properties and regression models. Journal of Statistical Distributions and Applications, 5(1), 7.

Altun, E., Yousof, H. M. and Hamedani, G. G. (2018d). A new log-location regression model with influence diagnostics and residual analysis. Facta Universitatis, Series: Mathematics and Informatics, 33(3), 417-449.

Altun, E., Yousof, H. M. and Hamedani, G. G. (2021b). The Gudermannian generated family of distributions with characterizations, regression models and applications. Studia Scientiarum Mathematicarum Hungarica, forthcoming.

Altun, E., Yousof, H. M., Chakraborty, S. and Handique, L. (2018e). Zografos-Balakrishnan Burr XII distribution: regression modeling and applications. International Journal of Mathematics and Statistics, 19(3), 46-70.

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(1), 126-141.

Balkema, A.A. and de Hann, L. Residual life at great age, Annals of Probability 2, 972–804, 1974.

Bryson, M. C. (1974). Heavy-tailed distribution: properties and tests. Technometrics 16, 161-68.

Chahkandi, M. and Ganjali, M. (2009). On some lifetime distributions with decrasing failure rate. Computational Statistics and Data Analysis 53, 4433–4440.

Chesneau, C. and Yousof, H. M. (2021). On a special generalized mixture class of probabilitic models. Journal of Nonlinear Modeling and Analysis, 3(1), 71-92.

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.

Durbey, S. D. (1970). Compound gamma, beta and F distributions. Metrika 16, 27–31.

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. https://doi.org/10.18187/pjsor.v16i4.3260

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. https://doi.org/10.19139/soic-2310-5070-1036

Gad, A. M., Hamedani, G. G., Salehabadi, S. M. and Yousof, H. M. (2019). The Burr XII-Burr XII distribution: mathematical properties and characterizations. Pakistan Journal of Statistics, 35(3), 229-248.

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.

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.

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.

Corbellini, A., Crosato, L., Ganugi, P and Mazzoli, M. (2007). Fitting Pareto II distributions on firm size: Statistical methodology and economic puzzles. Paper presented at the International Conference on Applied Stochastic Models and Data Analysis, Chania, Crete.

Cordeiro, G. M., Ortega, E. M. and Popovic, B. V. (2015). The gamma-Lomax distribution. Journal of Statistical computation and Simulation, 85(2), 305-319.

Elbiely, M. M. and Yousof, H. M. (2018). A new extension of the Lomax distribution and its Applications. Journal of Statistics and Applications, 2(1), 18-34.

Harris, C.M. (1968). The Pareto distribution as a queue service descipline. Operations Research, 16, 307–313.

Hassan, A.S. and Al-Ghamdi, A.S. (2009). Optimum step stress accelerated life testing for Lomax distibution. Journal of Applied Sciences Research, 5, 2153–2164.

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., Altun, E., Goual, H., and Yousof, H. M. (2020). 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.

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.

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.

Lemonte, A. J. and Cordeiro, G. M. (2013). An extended Lomax distribution. Statistics, 47(4), 800-816.

Lomax, K.S. (1954). Business failures: Another example of the analysis of failure data. Journal of the American Statistical Association, 49, 847–852.

Gleaton, J. U. and Lynch, J.D. (2006). Properties of generalized loglogistic families of lifetime distributions. Journal of Probability and Statistical Science, 4, 51-64.

Goual, H. and Yousof, H. M. (2019). Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodness-of-fit test. Journal of Applied Statistics, 47(1), 1-32.

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. (2018a). A new extended G family of continuous distributions with mathematical properties, characterizations and regression modeling. Pakistan Journal of Statistics and Operation Research, 737-758.

Hamedani, G. G., Rasekhi, M., Najibi, S., Yousof, H. M. and Alizadeh, M. (2019). Type II general exponential class of distributions. Pakistan Journal of Statistics and Operation Research, 15(2), 503-523.

Hamedani, G. G., Yousof, H. M., Rasekhi, M., Alizadeh, M. and Najibi, S. M., (2018b). Type I general exponential class of distributions. Pakistan Journal of Statistics and Operation Research, 14(1), 39-55.

Khalil, M. G., Hamedani, G. G. and 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.

Kotz, S. and Johnson, N. L. (1992). Breakthroughs in Statistics: Foundations and basic theory. Springer,Volume 1.

Kotz, S. and Nadarajah, S. (2000). Extreme value distributions: theory and applications. Imperial College Press, London.

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 mode lwith mathematical properties, copula, modified goodness-of-fit test for validation and real data modeling. Mathematics, 8(9), 1508.

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

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.

Mansour, M., Rasekhi, M., Ibrahim, M., Aidi, K., Yousof, H. M. and Elrazik, E. A. (2020d). A new parametric life distribution with modified Bagdonaviˇ cius-Nikulin goodness-of-fit test for censored validation, properties, applications, and different estimation methods. Entropy, 22(5), 592.

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.

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.

Murthy, D.N.P. Xie, M. and Jiang, R. (2004). Weibull Models, Wiley.

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.

Rodriguez-Lallena, J.A. and Ubeda-Flores, M. (2004). A new class of bivariate copulas. Statistics and Probability Letters, 66,315–25.

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.

Tadikamalla, P. R. (1980). A look at the Burr and realted distributions. International Statistical Review 48, 337–344.

Yadav, A. S., Altun, E. and Yousof, H. M. (2019). Burr-Hatke exponential distribution: a decreasing failure rate model, statistical inference and applications. Annals of Data Science, 1-20.

Tahir, M. H., Cordeiro, G. M., Mansoor, M. and Zubair, M. (2015). The Weibull-Lomax distribution: properties and applications. Hacettepe Journal of Mathematics and Statistics, 44(2), 461-480.

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.

Yousof, H. M., Ahsanullah, M. and 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.

Yousof, H. M., Ali, M. M., Goual, H. and Ibrahim. M. (2021). A new reciprocal Rayleigh extension: properties, copulas, different methods of estimation and modified right censored test for validation. Statistics in Transition New Series, forthcoming.

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(4), 723-740.

Yousof, H. M., Altun, E. and Hamedani, G. G. (2018a). A new extension of Frechet distribution with regression models, residual analysis and characterizations. Journal of Data Science, 16(4), 743-770.

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(1), 264-286.

Yousof, H. M., Altun, E., Ramires, T. G., Alizadeh, M. and Rasekhi, M. (2018). A new family of distributions with properties, regression models and applications. Journal of Statistics and Management Systems, 21, 1, 163-188.

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.

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.

*Statistics, Optimization & Information Computing*,

*9*(3), 555-586. https://doi.org/10.19139/soic-2310-5070-1101

- 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).