A Novel Family of Compound Probability Distributions: Properties, Copulas, Risk Analysis and Assessment under a Reinsurance Revenues Data Set

Abdelrahman  Mahmoud Khedr; Zohdy  Mohammed Nofal; Yehia  Mousa El Gebaly; Haitham  Mosad Yousof

Authors

Abdelrahman Mahmoud Khedr Department of Statistics, Mathematics and Insurance, Benha University, Benha, Egypt
Zohdy Mohammed Nofal Department of Statistics, Mathematics and Insurance, Benha University, Benha, Egypt
Yehia Mousa El Gebaly Department of Statistics, Mathematics and Insurance, Benha University, Benha, Egypt
Haitham Mosad Yousof Department of Statistics, Mathematics and Insurance, Benha University, Benha, Egypt

Keywords:

Ali-Mikhail-Haq copula, Farlie Gumbel Morgenstern copula, moments, reinsurance revenues data, risk analysis, zero truncated Poisson distribution

Abstract

It is possible that the probability-based distributions can effectively explain risk exposure. The level of risk exposure is typically expressed as one number, or at the very least, a small number of numbers. These risk exposure numbers, which unquestionably represent the results of a particular model, are commonly referred to as crucial critical risk indicators. Value-at-risk, tail-value-at-risk, tail variance, and tail mean-variance were the other four major risk indicators that were utilized to describe the risk exposure in the reinsurance revenues data. This paper gives a novel distribution for this use since these measurements were made using the suggested model. In this work, first we introduced a new compound G family with a strong physical motivation. Various structural mathematical and statistical properties are derived. The new density can be “asymmetric right skewed with heavy tail”, “symmetric” and bimodal density with different shapes. The new hazard function can be “upside-down-constant”, “increasing-constant”, “upside-down-increasing”, “increasing”, “decreasing” and “constant”. Many bivariate types have been also derived via different copulas. The estimation of the model parameters is performed by maximum likelihood method. Second, the usefulness and flexibility of the new family is illustrated by an application to real data set. Finally, the reinsurance revenues dataset’s risk level is examined using five major risk indicators. Along with an application, pertinent numerical analyses, and plots are included. There are certain noticeable and highlighted useful results.

References

Abouelmagd THM, Hamed MS, Hamedani GG, Ali MM, Goual H, Korkmaz MC, Yousof HM. The zero truncated Poisson Burr X family of distributions with properties, characterizations, applications, and validation test. J Nonlinear Sci App. 2019a; 12(5): 314-336.

Abouelmagd THM, Hamed MS, Handique L, Goual H, Ali MM, Yousof HM, Korkmaz MC. A new class of distributions based on the zero truncated Poisson distribution with properties and applications. J Nonlinear Sci App. 2019b; 12(3): 152-164.

Acerbi C, Tasche D. On the coherence of expected shortfall. J Bank Financ. 2002;26(7):1487-1503.

Ahmed B, Yousof HM. A new group acceptance sampling plans based on percentiles for the Weibull Fréchet Model. Stat Optim Inf Comput. 2023; 11(2): 409-421.

Ahmed B, Ali MM, Yousof HM. A novel G family for single acceptance sampling plan with application in quality and risk decisions. Ann Data Sci. 2024; 11: 181-199.

Al-babtain AA, Elbatal I, Yousof HM. A new flexible three-parameter model: properties, Clayton copula, and modeling real data. Symmetry. 2020; 12(3): 440; http://doi.org10.3390/

sym12030440

Ali MM, Mikhail NN, Haq MS. A class of bivariate distributions including the bivariate logistic. J Multivar Anal. 1978; 8(3): 405-412.

Alizadeh M, Afshari M, Contreras-Reyes JE, Mazarei D, Yousof HM. The extended Gompertz model: applications, mean of order P assessment, and statistical threshold risk analysis based on extreme stresses data. IEEE Trans Reliab. 2024; 1-13.

Aljadani A, Mansour MM, Yousof HM. A novel model for finance and reliability applications: theory, practices and financial peaks over a random threshold value-at-risk analysis. Pak J Stat Oper Res. 2024; 20(3): 489-515.

Almamy JA, Ibrahim M, Eliwa MS, Al-mualim S, Yousof HM. The two-parameter odd Lindley Weibull lifetime model with properties and applications. Int J Stat Prob. 2018; 7(4): 1927-7040.

Artzner P. Application of coherent risk measures to capital requirements in insurance. N Am Actuar J. 1999; 3(2): 11-25.

Aryal GR, Yousof HM. The exponentiated generalized-G Poisson family of distributions. Stochastics and Qual Control. 2017; 32(1): 1-17.

Cancho VG, Louzada-Neto F, Barriga GD. The Poisson-exponential lifetime distribution. Comput Stat Data Anal. 2011; 55(1): 677-686.

Chakraborty S, Handique L. The generalized Marshall-Olkin-Kumaraswamy-G family of distributions. J Data Sci. 2017; 15(3): 391-422.

Cordeiro GM, Ortega EMM, Nadarajah S. The Kumaraswamy Weibull distribution with application to failure data. J Franklin Inst. 2010; 347(8): 1399-1429.

Cordeiro GM, Yousof HM, Ramires TG, Ortega EM. The Burr XII system of densities: properties, regression model and applications. J Stat Comput Simul. 2018; 88(3): 432-456.

Dara ST, Ahmad M. Recent advances in moment distributions and their hazard rate. PhD dissertation. National College of Business Administration and Economics, Lahore, Pakistan; 2012.

Das J, Hazarika PJ, Alizadeh M, Contreras-Reyes JE, Mohammad HH, Yousof HM. Economic peaks and value-at-risk analysis: a novel approach using the Laplace distribution for house prices. Math Comput Appl. 2025; 30(1): 4; https://doi.org/10.3390/mca30010004

Elgohari H, Yousof HM. A generalization of Lomax distribution with properties, copula, and real data applications. Pak J Stat Oper Res. 2020a; 16(4): 697-711.

Elgohari H, Yousof HM. New extension of Weibull distribution: copula, mathematical properties and data modeling. Stat Optim Inf Comput. 2020b; 8(4): 972-993.

Elgohari H, Yousof HM. A new extreme value model with different copula, statistical properties and applications. Pak J Stat Oper Res. 2021; 17(4): 1015-1035.

Elgohari H, Ibrahim M, Yousof HM. A new probability distribution for modeling failure and service times: properties, copulas, and various estimation methods. Stat Optim Inf Comput. 2021; 9(3): 555-586.

El-Morshedy M, Eliwa MS, Al-Bossly A, Yousof HM. A new probability heavy-tail model for stochastic modeling under engineering data. J Math. 2022; 1910909: 1-20.

Elsayed HAH, Yousof HM. Extended Poisson generalized Burr XII distribution. J Appl Prob Stat. 2021; 16(1): 1-30.

Furman E, Landsman Z. Tail variance premium with applications for elliptical portfolio of risks. Astin Bull. 2006; 36(2): 433-462.

George R, Thobias S. Kumaraswamy Marshall-Olkin exponential distribution. Commun Stat Theory Methods. 2019; 48(8): 1920-1937.

Ghitany ME, Al-Hussaini EK, Al-Jarallah RA. Marshall-Olkin extended Weibull distribution and its application to censored data. J Appl Stat. 2005; 32(10): 1025-1034.

Goual H, Yousof HM. Validation of Burr XII inverse Rayleigh model via a modified chi-squared goodness-of-fit test. J Appl Stat. 2019; 47(3): 393-423.

Goual H, Yousof HM, Ali MM. Lomax inverse Weibull model: properties, applications, and a modified Chi-squared goodness-of-fit test for validation. J Nonlinear Sci App. 2020;13(6): 330-353.

Goual H, Yousof HM, Ali MM. Validation of the odd Lindley exponentiated exponential by a modified goodness of fit test with applications to censored and complete data. Pak J Stat Oper Res. 2019; 15(3): 745-771.

Gross J, Clark VA. Survival distributions: reliability applications in the biometrical sciences. New York: John Wiley and Sons; 1975.

Hamed MS, Cordeiro GM, Yousof HM. A new compound Lomax model: properties, copulas, modeling and risk analysis utilizing the negatively skewed insurance claims data. Pak J Stat Oper Res. 2022; 18(3): 601-631.

Hamedani GG, Altun E, Korkmaz MÇ, Yousof HM, Butt NS. A new extended G family of continuous distributions with mathematical properties, characterizations and regression modeling. Pak J Stat Oper Res. 2018; 14(3): 737-758.

Ibrahim M, Altun E, Yousof HM. A new distribution for modeling lifetime data with different methods of estimation and censored regression modeling. Stat Optim Inf Comput. 2020; 8(2): 610-630.

Ibrahim M, Hamedani GG, Butt NS, Yousof HM. Expanding the Nadarajah Haghighi model: copula, censored and uncensored validation, characterizations and applications. Pak J Stat Oper Res. 2022;18(3):537-553.

Ibrahim M, Yadav AS, Yousof HM, Goual H, Hamedani GG. A new extension of Lindley distribution: modified validation test, characterizations and different methods of estimation. Commun Stat Appl Methods. 2019; 26(5) :473-495.

Khan MI, Aljadani A, Mansour MM, Abd Elrazik EM, Hamedani GG, Yousof HM, Shehata WA. A new heavy-tailed Lomax model with characterizations, applications, peaks over random threshold value-at-risk, and the mean-of-order-p analysis. J Math. 2024; 2024(1): 5329529.

Korkmaz MÇ, Cordeiro GM, Yousof HM, Pescim RR, Afify AZ, Nadarajah S. The Weibull Marshall–Olkin family: Regression model and application to censored data. Commun Stat Theory Methods. 2018b; 48(16): 4171-4194.

Korkmaz MÇ, Yousof HM, Hamedani GG. The exponential Lindley odd log-logistic-G family: Properties, characterizations and applications. J Stat Theory Appl. 2018c;17(3):554-571.

Korkmaz MÇ, Yousof HM, Hamedani GG, Ali MM. The Marshall-Olkin Generalized G Poisson Family of Distributions. Pak J Stat. 2018d; 34(3): 251-267.

Lak F, Basikhasteh M, Alizadeh M, Yousof HM. The odd Log-Logistic generalized half-Normal lifetime Poisson model. Pak J Stat Oper Res. 2019; 15(1): 111-128.

Landsman Z. On the tail mean-variance optimal portfolio selection. Insur Math Econ. 2010; 46(3): 547-553.

Lee C, Famoye F, Olumolade O. Beta-Weibull distribution: some properties and applications to censored data. J Mod Appl Stat Methods. 2007; 6(1): 173-186.

Mansour MM, Butt NS, Ansari SI, Yousof HM, Ali MM, Ibrahim M. A new exponentiated Weibull distribution's extension: copula, mathematical properties and applications. Contributions to Mathematics. 2020a; 1: 57-66.

Mansour M, Korkmaz MC, Ali MM, Yousof HM, Ansari SI, Ibrahim M. A generalization of the exponentiated Weibull model with properties, Copula and application. Eurasian Bull Math. 2020b; 3(2): 84-102.

Mansour M, Rasekhi M, Ibrahim M, Aidi K, Yousof HM, Elrazik EA. A new parametric life distribution with modified Bagdonavičius-Nikulin goodness-of-fit test for censored validation, properties, applications, and different estimation methods. Entropy. 2020c; 22(5): 592.

Maurya SK, Nadarajah S. Poisson generated family of distributions: a review. Sankhya B. 2021; 83: 484-540.

Merovci F, Alizadeh M, Yousof HM, Hamedani GG. The exponentiated transmuted-G family of distributions: theory and applications. Commun Stat Theory Methods. 2017; 46(21): 10800-10822.

Merovci F, Yousof HM, Hamedani GG. The Poisson Topp Leone generator of distributions for lifetime data: theory, characterizations and applications. Pak J Stat Oper Res. 2020;16(2):343-355.

Mohamed HS, Ali MM, Yousof HM. The Lindley Gompertz model for estimating the survival rates: properties and applications in insurance. Ann Data Sci. 2023; 10: 1199-1216.

Mohamed HS, Cordeiro GM, Minkah R, Yousof HM, Ibrahim M. A size-of-loss model for the negatively skewed insurance claims data: applications, risk analysis using different methods and statistical forecasting. J Appl Stat. 2024; 51(2): 348-369.

Mohamed HS, Cordeiro GM, Yousof HM. The synthetic autoregressive model for the insurance claims payment data: modeling and future prediction. Stat Optim Inf Comput. 2025;in press.

Nasir A, Yousof HM, Jamal F, Korkmaz MÇ. The exponentiated Burr XII power series distribution: properties and applications. Stats. 2019; 2(1): 15-31.

Ramos MWA, Marinho PRD, Cordeiro GM, da Silva RV, Hamedani GG. The Kumaraswamy-G Poisson family of distributions. J Stat Theory Appl. 2015; 14(3): 222-239.

Rodriguez-Lallena JA, Ubeda-Flores MA. A new class of bivariate copulas. Stat Prob Lett. 2004; 66(3): 315-325.

Saber MM, Hamedani GG, Yousof HM, Butt NS, Ahmed B, Yousof HM. A family of continuous probability distributions: theory, characterizations, properties and different copulas. In: Ali MM, Ali I, Yousof HM, Ahmed MIM, editors. G Families of Probability Distributions. CRC Press, Taylor & Francis Group; 2023.

Saber MM, Mohie El-Din MM, Yousof HM. Reliability estimation for the remained stress-strength model under the generalized exponential lifetime distribution. J Prob Stat. 2021; 2021(1): 7363449.

Saber MM, Rasekhi M, Yousof HM. Generalized Stress-Strength and Generalized Multicomponent Stress-Strength Models. Stat Optim Inf Comput. 2022; in press.

Saber MM, Yousof HM. Bayesian and classical inference for generalized stress-strength parameter under generalized Logistic distribution. Stat Optim Inf Comput. 2023;11(3): 541-553.

Sen S, Korkmaz MC, Yousof HM. The quasi xgamma-Poisson distribution. Istatistik J Turk Stat Assoc. 2018a; 11(3): 65-76.

Sen S, Korkmaz MC, Yousof HM. The quasi xgamma-Poisson distribution: properties and application. Istatistik J Turk Stat Assoc. 2018b; 11(3): 65-76.

Shehata WAM, Aljadani A, Mansour MM, Alrweili H, Hamed MS, Yousof HM. A novel reciprocal-Weibull model for extreme reliability data: statistical properties, reliability applications, reliability port-var and mean of order p risk analysis. Pak J Stat Oper Res. 2024; 20(4): 693-718.

Tasche D. Expected shortfall and beyond. J Bank Financ. 2002; 26(7): 1519-1533.

Wirch JL. Raising Value at Risk. N Am Actuar J. 1999; 3(2): 106-115.

Yadav AS, Goual H, Alotaibi RM, Rezk H, Ali MM, Yousof HM. Validation of the Topp-Leone-Lomax model via a modified Nikulin-Rao-Robson goodness-of-fit test with different methods of estimation. Symmetry. 2020; 12(1): 57.

Yousof HM, Afify AZ, Alizadeh M, Hamedani GG, Jahanshahi SMA, Ghosh I. The generalized transmuted Poisson-G family of distributions. Pak J Stat Oper Res. 2018a; 14(4): 759-779.

Yousof HM, Afify AZ, Cordeiro GM, Alzaatreh A, Ahsanullah M. A new four-parameter Weibull model for lifetime data. J Stat Theory Appl. 2017a; 16(4): 448-466.

Yousof HM, Afify AZ, Hamedani GG, Aryal G. The Burr X generator of distributions for lifetime data. J Stat Theory Appl. 2017b; 16(3): 288-305.

Yousof HM, Aljadani A, Mansour MM, Abd Elrazik EM. A new Pareto model: risk application, reliability MOOP and port value-at-risk analysis. Pak J Stat Oper Res. 2024; 20(3): 383-407.

Yousof HM, Altun E, Ramires TG, Alizadeh M, Rasekhi M. A new family of distributions with properties, regression models and applications. J Stat Manag Syst. 2018b; 21(1): 163-188.

Yousof HM, Mansoor M, Alizadeh M, Afify AZ, Ghosh I. The Weibull-G Poisson family for analyzing lifetime data. Pak J Stat Oper Res. 2020; 16(1): 131-148.