A new losses (revenues) probability model with entropy analysis, applications and case studies for value-at-risk modeling and mean of order-P analysis

Ibrahim Elbatal; L. S. Diab; Anis Ben Ghorbal; Haitham M. Yousof; Mohammed Elgarhy; Emadeldin I. A. Ali; Ibrahim Elbatal; L. S. Diab; Anis Ben Ghorbal; Haitham M. Yousof; Mohammed Elgarhy; Emadeldin I. A. Ali

doi:10.3934/math.2024350

AIMS Mathematics

2024, Volume 9, Issue 3: 7169-7211. doi: 10.3934/math.2024350

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A new losses (revenues) probability model with entropy analysis, applications and case studies for value-at-risk modeling and mean of order-P analysis

1.
Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
2.
Department of Statistics, Mathematics and Insurance, Benha University, Benha, Egypt
3.
Department of Basic Sciences, Higher Institute of Administrative Sciences, Belbeis, AlSharkia, Egypt
4.
Mathematics and Computer Science Department, Faculty of Science, Beni-Suef University, Beni-Suef 62521, Egypt
5.
Department of Economics, College of Economics and Administrative Sciences, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
6.
Department of Mathematics, Statistics, and Insurance, Faculty of Business, Ain Shams University, Egypt

Received: 15 November 2023 Revised: 20 January 2024 Accepted: 25 January 2024 Published: 20 February 2024
MSC : 62F15, 62G20, 65C60

This study introduces the Inverse Burr-X Burr-XII (IBXBXII) distribution as a novel approach for handling asymmetric-bimodal claims and revenues. It explores the distribution's statistical properties and evaluates its performance in three contexts. The analysis includes assessing entropy, highlighting the distribution's significance in various fields, and comparing it to rival distributions using practical examples. The IBXBXII model is then applied to analyze risk indicators in actuarial data, focusing on bimodal insurance claims and income. Simulation analysis shows its preference for right-skewed data, making it suitable for mathematical modeling and actuarial risk assessments. The study emphasizes the IBXBXII model's versatility and effectiveness, suggesting it as a flexible framework for actuarial data analysis, particularly in cases of large samples and right-skewed data.
- asymmetric-bimodal claims data,
- asymmetric-bimodal insurance revenue data,
- losses (Gains) model,
- Tsallis entropy,
- value-at-risk,
- mean of order-P methodology
Citation: Ibrahim Elbatal, L. S. Diab, Anis Ben Ghorbal, Haitham M. Yousof, Mohammed Elgarhy, Emadeldin I. A. Ali. A new losses (revenues) probability model with entropy analysis, applications and case studies for value-at-risk modeling and mean of order-P analysis[J]. AIMS Mathematics, 2024, 9(3): 7169-7211. doi: 10.3934/math.2024350

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Abstract

This study introduces the Inverse Burr-X Burr-XII (IBXBXII) distribution as a novel approach for handling asymmetric-bimodal claims and revenues. It explores the distribution's statistical properties and evaluates its performance in three contexts. The analysis includes assessing entropy, highlighting the distribution's significance in various fields, and comparing it to rival distributions using practical examples. The IBXBXII model is then applied to analyze risk indicators in actuarial data, focusing on bimodal insurance claims and income. Simulation analysis shows its preference for right-skewed data, making it suitable for mathematical modeling and actuarial risk assessments. The study emphasizes the IBXBXII model's versatility and effectiveness, suggesting it as a flexible framework for actuarial data analysis, particularly in cases of large samples and right-skewed data.

References

[1]	I. W. Burr, Cumulative frequency functions, Ann. Math. Stat., 13 (1942), 215–232. https://doi.org/10.1214/aoms/1177731607 doi: 10.1214/aoms/1177731607
[2]	I. W. Burr, On a general system of distributions, III. The simple range, J. Am. Stat. Assoc., 63 (1968), 636–643.
[3]	I. W. Burr, P. J. Cislak, On a general system of distributions: I. Its curve-shaped characteristics; II. The sample median, J. Am. Stat. Assoc., 63 (1968), 627–635. https://doi.org/10.1080/01621459.1968.11009281 doi: 10.1080/01621459.1968.11009281
[4]	I. W. Burr, Parameters for a general system of distributions to match a grid of 3 and 4, Commun. Stat., 2 (1973), 1–21.
[5]	R. N. Rodriguez, A guide to the Burr-type XII distributions, Biometrika, 64 (1977), 129–134. https://doi.org/10.1093/biomet/64.1.129 doi: 10.1093/biomet/64.1.129
[6]	P. R. Tadikamalla, A look at the Burr and related distributions, Int. Stat. Rev., 48 (1980), 337–344. https://doi.org/10.2307/1402945 doi: 10.2307/1402945
[7]	P. F. Paranaíba, E. M. Ortega, G. M. Cordeiro, R. R. Pescim, The beta Burr XII distribution with application to lifetime data, Comput. Stat. Data An., 55 (2011), 1118–1136. https://doi.org/10.1016/j.csda.2010.09.009 doi: 10.1016/j.csda.2010.09.009
[8]	P. F. Paranaiba, E. M. Ortega, G. M. Cordeiro, M. A. D. Pascoa, The Kumaraswamy Burr XII distribution: Theory and practice, J. Stat. Comput. Sim., 83 (2013), 2117–2143. https://doi.org/10.1080/00949655.2012.683003 doi: 10.1080/00949655.2012.683003
[9]	A. Y. Al-Saiari, L. A. Baharith, S. A. Mousa, Marshall-Olkin extended Burr type XII distribution, Int. J. Stat. Probab., 3 (2014), 78–84. https://doi.org/10.5539/ijsp.v3n1p78 doi: 10.5539/ijsp.v3n1p78
[10]	N. Alsadat, V. B. Nagarjuna, A. S. Hassan, M. Elgarhy, H. Ahmad, E. M. Almetwally, Marshall-Olkin Weibull-Burr XII distribution with application to physics data, AIP Adv., 13 (2023). https://doi.org/10.1063/5.0172143
[11]	M. A. Zayed, A. S. Hassan, E. M. Almetwally, A. M. Aboalkhair, A. H. Al-Nefaie, H. M. Almongy, A compound class of unit Burr XII model: Theory, estimation, fuzzy, and application, Sci. Program., 2023 (2023). https://doi.org/10.1155/2023/4509889
[12]	A. Fayomi, A. S. Hassan, H. Baaqeel, E. M. Almetwally, Bayesian inference and data analysis of the unit-power Burr X distribution, Axioms, 12 (2023), 297. https://doi.org/10.3390/axioms12030297 doi: 10.3390/axioms12030297
[13]	A. S. Hassan, E. M. Almetwally, S. C. Gamoura, A. S. Metwally, Inverse exponentiated Lomax power series distribution: Model, estimation, and application, J. Math., 2022 (2022). https://doi.org/10.1155/2022/1998653
[14]	A. G. Abubakari, L. Anzagra, S. Nasiru, Chen Burr-Hatke exponential distribution: Properties, regressions and biomedical applications, Comput. J. Math. Stat. Sci., 2 (2023), 80–105. https://doi.org/10.21608/cjmss.2023.190993.1003 doi: 10.21608/cjmss.2023.190993.1003
[15]	H. M. Yousof, A. Z. Afify, G. G. Hamedani, G. Aryal, The Burr X generator of distributions for lifetime data, J. Stat. Theory Appl., 16 (2017), 288–305. https://doi.org/10.2991/jsta.2017.16.3.2 doi: 10.2991/jsta.2017.16.3.2
[16]	A. Z. Afify, G. M. Cordeiro, N. A. Ibrahim, F. Jamal, M. Elgarhy, M. A. Nasir, The Marshall-Olkin odd Burr III-G family: Theory, estimation, and engineering applications, IEEE Access, 9 (2020), 4376–4387. https://doi.org/10.1109/ACCESS.2020.3044156 doi: 10.1109/ACCESS.2020.3044156
[17]	R. A. Bantan, C. Chesneau, F. Jamal, I. Elbatal, M. Elgarhy, The truncated Burr X-G family of distributions: Properties and applications to actuarial and financial data, Entropy, 23 (2021), 1088. https://doi.org/10.3390/e23081088 doi: 10.3390/e23081088
[18]	M. Haq, M. Elgarhy, S. Hashmi, The generalized odd Burr III family of distributions: Properties, and applications, J. Taibah Univ. Sci., 13 (2019), 961–971. https://doi.org/10.1080/16583655.2019.1666785 doi: 10.1080/16583655.2019.1666785
[19]	S. K. Ocloo, L. Brew, S. Nasiru, B. Odoi, On the extension of the Burr XII distribution: Applications and regression, Comput. J. Math. Stat. Sci., 2 (2023), 1–30. https://doi.org/10.21608/cjmss.2023.181739.1000 doi: 10.21608/cjmss.2023.181739.1000
[20]	M. H. O. Hassan, I. Elbatal, A. H. Al-Nefaie, M. Elgarhy, On the Kavya-Manoharan-Burr X model: Estimations under ranked set sampling and applications, J. Risk Financ. Manag., 16 (2023), 19. https://doi.org/10.3390/jrfm16010019 doi: 10.3390/jrfm16010019
[21]	T. Bjerkedal, Acquisition of resistance in Guinea pigs infected with different doses of virulent tubercle bacilli, Am. J. Hyg., 72 (1960), 130–148.
[22]	G. M. Cordeiro, H. M. Yousof, T. G. Ramires, E. M. M. Ortega, The Burr XII system of densities: Properties, regression model and applications, J. Stat. Comput. Sim., 88 (2018), 432–456. https://doi.org/10.1080/00949655.2017.1392524 doi: 10.1080/00949655.2017.1392524
[23]	F. Figueiredo, M. I. Gomes, L. Henriques-Rodrigues, Value-at-risk estimation and the PORT mean-of-order-p methodology, Revstat, 15 (2017), 187–204.
[24]	E. Furman, Z. Landsman, Tail variance premium with applications for elliptical portfolio of risks, ASTIN Bull. J. IAA, 36 (2006), 433–462. https://doi.org/10.2143/AST.36.2.2017929 doi: 10.2143/AST.36.2.2017929
[25]	J. Havrda, F. Charvat, Quantification method of classification processes: Concept of structural entropy, Kybernetika, 3 (1967), 30–35.
[26]	Z. Landsman, On the tail mean-variance optimal portfolio selection, Insur. Math. Econ., 46 (2010), 547–553. https://doi.org/10.1016/j.insmatheco.2010.02.001 doi: 10.1016/j.insmatheco.2010.02.001
[27]	M. D. Nichols, W. J. Padgett, A bootstrap control chart for Weibull percentiles, Qual. Reliab. Eng. Int., 22 (2006), 141–151. https://doi.org/10.1002/qre.691 doi: 10.1002/qre.691
[28]	A. Rényi, On measures of entropy and information, In: Proceedings of the 4th Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA, 30 (1960), 547–561.
[29]	D. Tasche, Expected shortfall and beyond, J. Bank. Financ., 26 (2002), 1519–1533. https://doi.org/10.1016/S0378-4266(02)00272-8 doi: 10.1016/S0378-4266(02)00272-8
[30]	C. Acerbi, D. Tasche, On the coherence of expected shortfall, J. Bank. Financ., 26 (2002), 1487–1503. https://doi.org/10.1016/S0378-4266(02)00283-2 doi: 10.1016/S0378-4266(02)00283-2
[31]	C. Tsallis, The role of constraints within generalized non-extensive statistics, Physica, 261 (1998), 547–561.
[32]	M. M. A. El-Raouf, M. A. Oud, A novel extension of generalized Rayleigh model with engineering applications, Alex. Eng. J., 73 (2023), 269–283. https://doi.org/10.1016/j.aej.2023.04.063 doi: 10.1016/j.aej.2023.04.063
[33]	R. Joshi, A new picture fuzzy information measure based on Tsallis-Havrda-Charvat concept with applications in presaging poll outcome, Comput. Appl. Math., 39 (2020), 71. https://doi.org/10.1007/s40314-020-1106-z doi: 10.1007/s40314-020-1106-z
[34]	J. Wirch, Raising value at risk, N. Am. Actuar. J., 3 (1999), 106–115. https://doi.org/10.1080/10920277.1999.10595804
[35]	R. Zhou, R. Cai, G. Tong, Applications of entropy in finance: A review, Entropy, 15 (2013), 4909–4931. https://doi.org/10.3390/e15114909 doi: 10.3390/e15114909
[36]	M. Ormos, D. Zibriczky, Entropy-based financial asset pricing, Plos One, 9 (2014), e115742. https://doi.org/10.1371/journal.pone.0115742 doi: 10.1371/journal.pone.0115742
[37]	R. Aloui, S. B. Jabeur, H. Rezgui, W. B. Arfi, Geopolitical risk and commodity future returns: Fresh insights from dynamic copula conditional value-at-risk approach, Resour. Policy, 85 (2023), 103873. https://doi.org/10.1016/j.resourpol.2023.103873 doi: 10.1016/j.resourpol.2023.103873
[38]	M. Bernardi, L. Catania, Comparison of value-at-risk models using the MCS approach, Comput. Stat., 31 (2016), 579–608. https://doi.org/10.1007/s00180-016-0646-6 doi: 10.1007/s00180-016-0646-6
[39]	Y. Dong, Z. Dong, An innovative approach to analyze financial contagion using causality-based complex network and value at risk, Electronics, 12 (2023), 1846. https://doi.org/10.3390/electronics12081846 doi: 10.3390/electronics12081846
[40]	M. I. Gomes, M. F. Brilhante, D. Pestana, A mean-of-order-p class of value-at-risk estimators, In: Theory and Practice of Risk Assessment: ICRA 5, Tomar, Portugal, Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-18029-8_23
[41]	C. Trucíos, J. W. Taylor, A comparison of methods for forecasting value at risk and expected shortfall of cryptocurrencies, J. Forecasting, 42 (2023), 989–1007. https://doi.org/10.1002/for.2929 doi: 10.1002/for.2929
[42]	Z. Zou, Q. Wu, Z. Xia, T. Hu, Adjusted Ré nyi entropic value-at-risk, Eur. J. Oper. Res., 306 (2023), 255–268. https://doi.org/10.1016/j.ejor.2022.08.028 doi: 10.1016/j.ejor.2022.08.028

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