# A Novel Two-parameter Nadarajah-Haghighi Extension: Properties, Copulas, Modeling Real Data and Different Estimation Methods

### Abstract

A new two-parameter lifetime distribution is proposed and numerically studied. The new model has a flflexible failure rate shapes such as “monotonically increasing” , “monotonically decreasing” , “bathtub” , “constant” , “upside down” and “J-shape” . Various of its statistical properties are derived. A numerical analysis of skewness and kurtosis are presented. Many bivariate and multivariate extensions are also presented via Farlie Gumbel Morgenstern copula, Renyi entropy copula, modifified Farlie Gumbel Morgenstern copula and Clayton copula. Several estimation methods such as the maximum likelihood, Cramer-von-Mises, L-moment estimation, Anderson Darling, right Tail-Anderson Darling estimation and left tail-Anderson Darling are presented and considered. Numerical simulations are performed to assess the performance of estimation methods. An environmental data set is employed to measure flflexibility of the new model also to compare the estimation methods.### References

Aarset, M. V. How to identify bathtub hazard rate. IEEE Transactions on Reliability, 36, pp. 106-108, 1987.

Ali, M. M., Yousof, H. M. and Ibrahim, M. 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, VOL 1, pp. 1-25, 2021a.

Ali, M. M., Ibrahim, M. and Yousof, H. M. 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, VOL 1, pp. 26-49, 2021b.

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

Alizadeh, M., Rasekhi, M., Yousof, H. M., Ramires, T. G. and Hamedani G. G. Extended exponentiated Nadarajah- Haghighi model: mathematical properties, characterizations and applications. Studia Scientiarum Mathematicarum Hungarica, 55, pp. 498–522, 2018.

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

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

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

Altun, E., Yousof, H. M. and Hamedani G. G. A Flexible Extension of Generalized Half-Normal Distribution: Characterizations and Regression Models. International Journal of Applied Mathematics and Statistics, 57(3), pp. 27-49, 2018d.

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

Choulakian, V., Stephens, M.A. Goodness-of-fit for the generalized Pareto distribution. Technometrics, 43, pp. 478–484, 2001.

Dias, C. R. B., Alizadeh, M. and Cordeiro, G. M. The beta Nadarajah-Haghighi distribution. Hacettepe Journal of Mathematics and Statistics, 47, pp. 1302-1323, 2018.

Elgohari, H. and Yousof, H. M. A Generalization of Lomax Distribution with Properties, Copula and Real Data Applications. Pakistan Journal of Statistics and Operation Research, 16(4), pp. 697-711, 2020a. https://doi.org/10.18187/pjsor.v16i4.3260

Elgohari, H. and Yousof, H. M. New Extension of Weibull Distribution: Copula, Mathematical Properties and Data Modeling. Statistics, Optimization & Information Computing, 8(4), pp. 972-993, 2020b. https://doi.org/10.19139/soic-2310-5070-1036

Elsayed, H. A. H. and Yousof, H. M. The Burr X Nadarajah Haghighi distribution: statistical properties and application to the exceedances of flood peaks data. Journal of Mathematics and Statistics, 15, pp. 146-157, 2019.

Gupta, R.D. and Kundu, D. Generalized exponential distributions. Aust. N. Z. J. Stat. 41, pp. 173–188, 1999.

Gumbel, E. J. Bivariate logistic distributions. Journal of the American Statistical Association, 56, pp. 335-349, 1961.

Gumbel, E. J. Bivariate exponential distributions. Journal of the American Statistical Association, 55, pp. 698-707, 1960.

Ibrahim, M. The generalized odd log-logistic Nadarajah Haghighi distribution: statistical properties and different methods of estimation. J. Appl. Probab. Stat, 15, pp. 61-84, 2020.

Lemonte, A. J. A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtub-shaped failure rate function. Computational Statistics & Data Analysis, 62, 149-170, 2013.

Lemonte, A. J., Cordeiro, G. M. and Moreno–Arenas, G. A new useful three-parameter extension of the exponential distribution. Statistics, 50(2), pp. 312-337, 2016.

MacDonald, P. D. M. Comment on An estimation procedure for mixtures of distributions by Choi and Bulgren. Journal of the Royal Statistical Society. Series B, 33(2), pp. 326-329, 1971.

Mansour, M. M., Ibrahim, M., Aidi, K., Shafique Butt, N., Ali, M. M., Yousof, H. M. and Hamed, M. S. A New Log-Logistic Lifetime Model with Mathematical Properties, Copula, Modified Goodness-of-Fit Test for Validation and Real Data Modeling. Mathematics, 8(9), pp. 1508, 2020a.

Mansour, M. M., Butt, N. S., Ansari, S. I., Yousof, H. M., Ali, M. M. and Ibrahim, M. A new exponentiated Weibull distribution’s extension: copula, mathematical properties and applications. Contributions to Mathematics, 1 (2020), pp. 57–66, 2020b. DOI: 10.47443/cm.2020.0018

Mansour, M., Korkmaz, M. C., Ali, M. M., Yousof, H. M., Ansari, S. I. and Ibrahim, M. A generalization of the exponentiated Weibull model with properties, Copula and application. Eurasian Bulletin of Mathematics, 3(2), pp. 84-102, 2020c.

Mansour, M., Rasekhi, M., Ibrahim, M., Aidi, K., Yousof, H. M. and Elrazik, E. A. 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), pp. 592, 2020d.

Mansour, M., Yousof, H. M., Shehata,W. A. and Ibrahim, M. 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), pp. 223-238, 2020e.

Mansour, M. M., Butt, N. S., Yousof, H. M., Ansari, S. I. and Ibrahim, M. 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), pp. 373-386, 2020f.

Morgenstern, D. Einfache beispiele zweidimensionaler verteilungen. Mitteilingsblattfur Mathematische Statistik, 8, pp. 234-235, 1956.

Nadarajah, S.and Haghighi, F. An extension of the exponential distribution. Statistics. 45, pp. 543–558, 2011.

Nascimento, A. D., Silva, K. F., Cordeiro, G. M., Alizadeh, M., Yousof, H. M. and Hamedani, G. G. The odd Nadarajah-Haghighi family of distributions: properties and applications. Studia Scientiarum Mathematicarum Hungarica, 56(2), pp. 185-210, 2019.

Ortega, E. M., Lemonte, A. J., Silva, G. O. and Cordeiro, G. M. New flexible models generated by gamma random variables for lifetime modeling. Journal of Applied Statistics, 42(10), pp. 2159-2179, 2015.

Pougaza, D. B. and Djafari, M. A. Maximum entropies copulas. Proceedings of the 30th international workshop on Bayesian inference and maximum. Entropy methods in Science and Engineering, pp. 329-336, 2011.

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

Renyi, A. On measures of entropy and information. In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability I, University of California Press, Berkeley, pp. 547-561, 1961.

Shannon, C.E. A mathematical theory of communication. Bell System Technical Journal. 27, 379-432, 1948.

Silva, F. S., Percontini, A., E., Ramos, de Brito, M.W., Venancio, R. and Cordeiro, G. M. The Odd Lindley-G Family of Distributions. Austrian Journal of Statistics, 46(1), pp. 65-87, 2017.

Yousof, H. M., Ali, M. M., Aidi, K., Hamedani, G. G. and Ibrahim, M. A new lifetime distribution with properties, characterizations, validation testing, different estimation methods. Statistics, Optimization & Information Computing, forthcoming 2021.

Yousof, H. M., Altun, E., Rasekhi, M., Alizadeh, M., Hamedani, G. G. and Ali, M. M. A new lifetime model with regression models, characterizations and applications. Communications in Statistics-Simulation and Computation, 48(1), pp. 264-286, 2019.

Yousof, H. M. and Korkmaz, M. C. Topp-Leone Nadarajah-Haghighi distribution: mathematical properties and applications, International Journal of Applied Mathematics. Journal of Statisticians: Statistics and Actuarial Sciences, 2, pp. 119-128, 2017.

Yousof, H. M., Hamedani, G. G. and Ibrahim, M. (2020). The Two-parameter Xgamma Fr´echet Distribution: Characterizations, Copulas, Mathematical Properties and Different Classical Estimation Methods. Contributions to Mathematics, 2, 32-41, 2020.

Yousof, H. M., Korkmaz, M. C. Hamedani G. G. The odd Lindley Nadarajah-Haghighi distribution. J. Math. Comput. Sci., 7, pp. 864-882, 2017.

*Statistics, Optimization & Information Computing*,

*10*(3), 725-749. https://doi.org/10.19139/soic-2310-5070-1250

- 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).