## Main Article Content

## Abstract

In this work, we introduced a new three-parameter Nadarajah-Haghighi model. We derived explicit expressions for some of it statistical properties. The Farlie Gumbel Morgenstern, modified Farlie Gumbel Morgenstern, Clayton, Renyi and Ali-Mikhail-Haq copulas are used for deriving some bivariate type extensions. We consider maximum likelihood, CramÃ©r-von-Mises, ordinary least squares, whighted least squares, Anderson Darling, right tail Anderson Darling and left tail Anderson Darling estimation procedures to estimate the unknown model parameters. Simulation study for comparing estimation methods is performed. An application for comparing methods as also presented. The maximum likelihood estimation method is the best method. However, the other methods performed well. Another application for comparing the competitive models is investigated.

## Keywords

## Article Details

This work is licensed under a Creative Commons Attribution 4.0 International License.

**Authors who publish with this journal agree to the following License**

**CC BY: **This license allows reusers to distribute, remix, adapt, and build upon the material in any medium or format, so long as attribution is given to the creator. The license allows for commercial use.

*How to Cite*

*Pakistan Journal of Statistics and Operation Research*,

*18*(2), 437-463. https://doi.org/10.18187/pjsor.v18i2.3938

* * References

- 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) https://doi.org/10.1007/s40840-022-01250-y
- 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.
- Ahmed, B., Ali, M. M. and Yousof, H. M. (2022). A Novel G Family for Single Acceptance Sampling Plan with Application in Quality and Risk Decisions, Annals of Data Science, forthcoming.
- 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.
- 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(3), 615-623.
- 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.
- 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.
- 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.
- 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(1), 145-150.â€
- 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.
- 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(4), 1-27.â€
- 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(3), 417-449.
- 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.
- 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(3), 46-70.
- Aryal, G. R. and Yousof, H. M. (2017). The exponentiated generalized-G Poisson family of distributions. Economic Quality Control, 32(1), 1-17.
- Balakrishnan, N., Basu, A.P. (1995). The exponential distribution. theory, methods and applications. Gordon and Breach, Amsterdam, The Netherlands.
- Chesneau, C., Yousof, H. M., Hamedani, G. and Ibrahim, M. (2022). A New One-parameter Discrete Distribution: The Discrete Inverse BurrDistribution: Characterizations, Properties, Applications, Bayesian and Non-Bayesian Estimations. Statistics, Optimization & Information Computing, 10(2), 352-371.
- Cordeiro, G. M., Gomes, A. E., da-Silva, C. Q. and Ortega, E. M. M. (2013). The beta exponentiated Weibull distribution. Journal of Statistical Computation and Simulation, 83, 114-138.
- Dias, C. R. B. (2016), New continuous distributions applied to lifetime data and survival analysis . Doctoral thesis, Universidade Federal de Pernumbuco, Recife.
- 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.
- 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.
- 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.
- 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.
- 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. 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.
- 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.
- 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., Najib, 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., Aidi, K., Ali, M. M. and Yousof, H. M. (2022). A Novel Test Statistic for Right Censored Validity under a new Chen extension with Applications in Reliability and Medicine. Annals of Data Science, forthcoming.
- Ibrahim, M., Ali, M. M. and Yousof, H. M. (2022). The discrete analogue of the Weibull G family: properties, different applications, Bayesian and non-Bayesian estimation methods. Annals of Data Science, forthcoming.
- 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.
- 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.
- 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(1), 121-161.â€
- Klein, J. P. and Moeschberger, M. L. (1997). Survival Analysis Techniques for Censored and Truncated Data. Springer Verlag, New York.
- Korkmaz, M. Ã‡., 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.â€
- 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(3), 554 - 571.
- 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(3), 251-267.
- Lemonte, A. J. (2013). 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.
- Lima, S. R. L. (2015), The half normal generalized family and kummaraswamy Nadarajah Haghighi distribution. Master thesis, Universidade Federal de Pernumbuco.
- 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.
- 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.
- Mansour, M., Korkmaz, M. C., Ali, M. M., Yousof, H. M., Ansari, S. I., & 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 BagdonaviÄius-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.
- Marshall, A.W., Olkin, I. (2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. In: Springer Series in Statistics, New York.
- 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(2), 343-355.â€
- 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.
- Mohamed, H. S., Ali, M. M. and Yousof, H. M. (2022). The Lindley Gompertz Model for Estimating the Survival Rates: Properties and Applications in Insurance, Annals of Data Science, forthcoming.
- Nadarajah, S. and Haghighi, F. (2011). An extension of the exponential distribution. Statistics: A Journal of Theoretical and Applied Statistics, 45, 543-558.
- 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.
- Ortega, E. M. M., Lemonte, A. J., Silva, G. O., and Cordeiro, G. M. (2015). New flexible models generated by gamma random variables for lifetime modeling. Journal of Applied Statistics, 42, 2159-2179.
- 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 neWClass of bivariate copulas. Statistics and Probability Letters, 66, 315-25.
- 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.
- Saber, M. M. Marwa M. Mohie El-Din and Yousof, H. M. (2022). Reliability estimation for the remained stress-strength model under the generalized exponential lifetime distribution, Journal of Probability and Statistics, 2021, 1-10.
- 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.
- 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.
- 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(4), 943-961.
- Tsuang, M. T. AND Woolson, R. F. (1977). Mortality in patients with schizophrenia, mania and depression. British Journal of Psychiatry, 130, 162-166.
- VedoVatto, T., Nascimento, A. D. C., Lima, M. C. S., Pinho, L. G. B. and Cordeiro, G. M. (2016). Some computational and theoretical aspects of the exponentiated generalized Nadarajah-Haghighi distribution. arXiv preprint arXiv:1610.08876.
- 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.
- 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(2), 457-483.
- 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(2), 519-547.
- Yousof, H. M., Alizadeh, M., Jahanshahi, 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. Many other new NH versions can be derived based on Yousof et al. (2017 and 2018)
- 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, 163-188.
- Yousof, H. M., Altun, E., Rasekhi, M., Alizadeh, M., Hamedani, G. G. and Ali, M. M. (2019). A new lifetime model with regression models, characterizations and applications. Communications in Statistics-Simulation and Computation, 48(1), 264-286.
- Yousof, H. M. and Korkmaz, M. C. (2017). Topp-Leone Nadarajah-Haghighi distribution: mathematical properties and applications, International Journal of Applied Mathematics. Journal of Statisticians: Statistics and Actuarial Sciences, 2, 119-128.
- Yousof, H. M., Chesneau, C., Hamedani, G. and Ibrahim, M. (2021). A New Discrete Distribution: Properties, Characterizations, Modeling Real Count Data, Bayesian and Non-Bayesian Estimations. Statistica, 81(2), 135-162.