# A Novel Two-Parameter Compound G Family of Probability Distributions With Some Copulas, Statistical Properties and Applications

### Abstract

In this work, we introduce a new G family with two-parameter called the compound reversed Rayleigh-G family. Several relevant mathematical and statistical properties are derived and analyzed. The new density can be heavy tail and right skewed with one peak, symmetric density, simple right skewed density with one peak, asymmetric right skewed with one peak and a heavy tail and right skewed with no peak. The new hazard function can be "upsidedown-constant", "constant", "increasing-constant", "revised J shape", "upside-down", "J shape" and "increasing". Many bivariate types have been also derived via di¤erent common copulas. The estimation of the model parameters is performed by maximum likelihood method. The usefulness and ‡exibility of the new family is illustrated by means of two real data sets.### References

Aarset, M. V. (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 36(1), 106-108.

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

Al-babtain, A. A., Elbatal, I. and Yousof, H. M. (2020). A New Flexible Three-Parameter Model: Properties, Clayton Copula, and Modeling Real Data. Symmetry, 12(3), 440.

Ali, M. M., Ibrahim, M. and Yousof, H. M. (2021a). Expanding the Burr of the type X model: properties, copula, real data modeling and different methods of estimation. Optimal Decision Making in Operations Research & Statistics: Methodologies and Applications, VOL 1, 21-42.

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, VOL 1, 1-20.

Almamy, J. A., Ibrahim , M., Eliwa, M. S., Al-mualim, S., & Yousof, H. M. (2018). The two-parameter odd Lindley Weibull lifetime model with properties and applications. International Journal of Statistics and Probability, 7(4), 1927- 7040.

Alizadeh, M., Yousof, H. M. Rasekhi, M. and Altun, E. (2019). The odd log-logistic Poisson-G Family of distributions, Journal of Mathematical Extensions, 12(3), 81-104.

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.

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.

Ahmed, B., Ali, M. M. and Yousof, H. M. (2022). A novel G family for single acceptance sampling pan 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.

Aryal, G. R. and Yousof, H. M. (2017). The exponentiated generalized-G Poisson family of distributions. Economic Quality Control, 32(1), 1-17.

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.

Cancho, V. G., Louzada-Neto, F. and Barriga, G. D. (2011). The Poisson-exponential lifetime distribution. Computational Statistics & Data Analysis, 55(1), 677-686.

Chesneau, C., Yousof, H. M., Hamedani, G. and Ibrahim, M. (2022). A New One-parameter Discrete Distribution: The Discrete Inverse Burr Distribution: Characterizations, Properties, Applications, Bayesian and Non-Bayesian Estimations. Statistics, Optimization & Information Computing, 10, 352-371.

Cordeiro, G. M., Yousof, H. M., Ramires, T. G. and Ortega, E. M. (2018). The Burr XII system of densities: properties,

regression model and applications. Journal of Statistical Computation and Simulation, 88(3), 432-456.

Dara, S.T. and Ahmad, M. (2012). Recent Advances in Moment Distributions and their Hazard Rate. Ph.D. Thesis. National College of Business Administration and Economics, Lahore, Pakistan.

Eugene, N., Lee, C. and Famoye, F. (2002). Beta-normal distribution and its applications. Commun. Stat. Theory Methods, 31, 497-512.

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-

-1036

El-Morshedy, M., Alshammari, F. S., Hamed, Y. S., Eliwa, M. S., Yousof, H. M. (2021). A New Family of Continuous Probability Distributions. Entropy, 23, 194. https://doi.org/10.3390/e23020194

Ghitany, M.E., Al-Hussaini, E.K., AlJarallah, R. A. (2005). Marshall–Olkin extended Weibull distribution and its application to censored data. J Appl Stat, 32:1025–1034.

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, 745-771.

Goual, H. and Yousof, H. M. (2020). Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodnessof- 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 (JNSA), 13(6), 330-353.

Gross, J. and Clark, V. A. (1975). Survival Distributions: Reliability Applications in the Biometrical Sciences, John

Wiley, New York, USA.

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., Ali, M. M. and Yousof, H. M. (2022b). The discrete analogue of the Weibull G family: properties, different

applications, Bayesian and non-Bayesian estimation methods. Annals of Data Science, forthcoming.

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 and Information Computing, 8(2), 610-630.

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.

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. and Yousof, H. M. (2019). The Burr of the type X exponentiated Weibull model: Characterizations, mathematical properties and applications to failure and survival times data. Pakistan Journal of Statistics and Operation Research, 141-160.

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,

(1), 121-161.

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,

(1), 59-74.

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.

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. 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 Bagdonavičius–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. and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the Exponential and Weibull families. Biometrika, 84, 641-652.

Maurya, S. K. and Nadarajah, S. (1998). Poisson Generated Family of Distributions: A Review. Sankhya B, 1-57.

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.

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.

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.

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.

Mohamed, H. S., Cordeiro, G. M. and Yousof, H. M. (2022). The synthetic autoregressive model for the insurance claims payment data: modeling and future prediction. Statistics, Optimization & Information Computing, forthcoming.

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.

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-

Rezaei, S., B. B. Sadr, M. Alizadeh, and S. Nadarajah. (2017). Topp-Leone generated family of distributions: Properties

and applications. Communications in Statistics: Theory and Methods 46 (6), 2893–2909.

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., Hamedani, G. G., Yousof, H. M. But, N. S., Ahmed, B. and Yousof, H. M. (2022) A Family of Continuous

Probability Distributions: Theory, Characterizations, Properties and Different Copulas, CRC Press, Taylor & Francis Group.

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.

Saber, M. M., Rasekhi, M. and Yousof, H. M. (2022). Generalized Stress-Strength and Generalized Multicomponent

Stress-Strength Models, Statistics, Optimization & Information Computing, forthcoming.

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.

Shehata, W. A. M. and Yousof, H. M. (2021). The four-parameter exponentiated Weibull model with copula, properties

and real data modeling. Pakistan Journal of Statistics and Operation Research, 17, 649-667.

Shehata, W. A. M. and Yousof, H. M. (2022). A novel two-parameter Nadarajah-Haghighi extension: properties, copulas, modeling real data and different estimation methods. Statistics, Optimization & Information Computing, 10, 725-749.

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, 249-272.

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, 943-961.

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., 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., Hamedani G. G., Jahanshahi, S. M. A., Ghosh, I., (2018a). The generalized

transmuted Poisson-G family of Distributions, Pak. J. Stat. Oper. Res., Vol. 14 (4), pp. 759-779.

Yousof, H. M., Afify, A. Z., Hamedani, G. G. and Aryal, G. (2017a). The Burr of the type X generator of distributions

for lifetime data. Journal of Statistical Theory and Applications, 16, 288–305.

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., Jahanshahi, S. M. A., Ramires, T. G., Ghosh, I. and Hamedani, G. G. (2017b). The

transmuted Topp-Leone G family of distributions: theory, characterizations and applications. Journal of Data Science,

(4), 723-740.

Yousof, H. M., Altun, E., Ramires, T. G., Alizadeh, M. and Rasekhi, M. (2018b). A new family of distributions with

properties, regression models and applications, Journal of Statistics and Management Systems, 21(1), 163-188.

Yousof, H. M., Chesneau, C., Hamedani, G. and Ibrahim, M. (2021c). A new discrete distribution: properties,

characterizations, modeling real count data, Bayesian and non-Bayesian estimations. Statistica, 81, 135-162.

Yousof, H. M., Mansoor, M. Alizadeh, M., Afify, A. Z., Ghosh, I. and Afify, A. Z. (2020). The Weibull-G Poisson family

for analyzing lifetime data. Pak. J. Stat. Oper. Res., 16 (1), 131-148.

*Statistics, Optimization & Information Computing*,

*11*(2), 345-367. https://doi.org/10.19139/soic-2310-5070-1436

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