Modeling asymmetrically dependent automobile bodily injury claim data using Khoudraji Copulas

Year 2024, Volume: 42 Issue: 4, 1183 - 1193, 01.08.2024

Emel Kızılok Kara , Sibel Açık Kemaloğlu

Abstract

Linear-dependent variables are typically modeled through the Spearman correlation, a classical statistical technique. In reality, the dependence between the data cannot always be linear. The copula approach has often been a popular tool for modeling dependent data in these cases. Archimedean copulas, which can model mostly symmetrical data, are also among the copula families used for this purpose. Recently, asymmetric copula models have been developed to model unsymmetrical-dependent variables. The dependency measure is calculated using directional dependency coefficients instead of the Spearman correlation when the data is asymmetrical. Appropriate asymmetric model selection is made with the help of these measurements.
In the study, first, dependency parameters corresponding to different Spearman coefficients were obtained for Archimedean copula families, and asymmetric copulas were derived from them. Then, simulation data were obtained for these parameter values to determine the effect of asymmetry on data modeling, and directional dependency measures were found. In addition, the study methodology was applied to automobile bodily injury claims data, which is a real dataset with an asymmetric structure. Here, we used two different asymmetric models: the Khoudraji copula KC models, which are created by multiplying independent and Archimedean copulas, and the LCC models, which are linear-convex combinations of Archimedean copulas. Finally, the appropriate model was selected according to the directional dependency coefficients, and the results were interpreted.

Keywords

Asymmetric Copula, Archimedean Copula, Automobile Bodily Injury Claims Data, Directional Dependence

References

• REFERENCES [1] Sklar A. Functions de repartition an dimensions at leurs marges. Publ Inst Statist Univ Paris 1959;8:229231.
• [2] Nelsen RB. An Introduction to Copulas. 2nd ed. New York: Springer; 2006.
• [3] Salvadori G, De Michele C. On the use of copulas in hydrology: theory and practice. J Hydrol Eng 2007;12:369–380. [CrossRef]
• [4] Genest C, Favre A-C. Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol Eng 2007;12:347–368. [CrossRef]
• [5] Joe H. Dependence Modeling with Copulas. Boca Rotan, Florida: CRC Press; 2014.
• [6] Durante F, Sempi C. Principles of copula theory. Boca Rotan, Florida: CRC Press; 2015. [CrossRef]
• [7] Hong Y, Wang JP, Li DQ, Cao ZJ, Ng CWW, Cui P. Statistical and probabilistic analyses of impact pressure and discharge of debris flow from 139 events during 1961 and 2000 at Jiangjia Ravine, China. Eng Geol 2015;187:122–134. [CrossRef]
• [8] Salvadori G, De Michele C, Kottegoda NT, Rosso R. Extremes in nature: an approach using copulas. Water Science and Technology Library. Berlin/Heidelberg: Springer Science &Business Media; 2017.
• [9] Kara EK, Yıldız O. Bivariate analysis of precipitation and using copulas. Istatistik J Turk Stat Assoc 2014;7:6370.
• [10] Callau Poduje AC, Belli A, Haberlandt UA. Dam risk assessment based on univariate versus bivariate statistical approaches: A case study for Argentina. Hydrol Sci J 2014;59:2216–2232. [CrossRef]
• [11] Kara EK. The earthquake risk analysis based on copula models for Turkey. Sigma J Eng Nat Sci 2017;35:187200.
• [12] Kim JM, Jung YS, Sungur EA. Truncation invariant copulas for modeling directional dependence: Application to foreign currency exchange data. Model Assist Stat Appl 2014;9:309324. [CrossRef]
• [13] Kemaloglu SA, Kara EK. Modeling dependent financial assets by dynamic copula and portfolio optimization based on CVaR. Commun Fac Sci Univ Ankara Ser A1 Math Stat 2015;64:113. [CrossRef]
• [14] Kara EK, Kemaloglu SA. Portfolio optimization of dynamic copula models for dependent financial data using change point approach. Commun Fac Sci Univ Ankara Ser A1 Math Stat 2016;65:175188. [CrossRef]
• [15] Kara EK, Kemaloglu SA, Evkaya ÖO. Modeling Currency Exchange Data with Asymmetric Copula Functions. In: Terzioğlu, M.K. (eds) Advances in Econometrics, Operational Research, Data Science and Actuarial Studies. Contributions to Economics. Springer, Cham., 2022, p. 4962. [CrossRef]
• [16] Moradian S, Olbert AI, Gharbia S, Iglesias G. Copula-based projections of wind power: Ireland as a case study. Renew Sustain Energy Rev 2023;175:113147. [CrossRef]
• [17] Corbella S, Stretch DD. Simulating a multivariate sea storm using Archimedean copulas. Coastal Eng 2013;76:68–78. [CrossRef]
• [18] Ma P, Zhang Y. Modeling asymmetrically dependent multivariate ocean data using truncated copulas. Ocean Eng 2022;244:110226. [CrossRef]
• [19] Li X, Babovic V. Multi-site multivariate downscaling of global climate model outputs: an integrated framework combining quantile mapping, stochastic weather generator and Empirical Copula approaches. Clim Dyn 2019;52:57755799. [CrossRef]
• [20] Kim JM, Jung YS, Sungur EA, Han KH, Park C, Sohn IA. Copula method for modeling directional dependence of genes. BMC Bioinform 2008;9:225. [CrossRef]
• [21] Kim JM, Jung YS, Soderberg T. Directional dependence of genes using survival truncated FGM type modification copulas. Commun Stat Simul Comput 2009;38:14701484. [CrossRef]
• [22] Sungur EA. A note on directional dependence in regression setting. Commun Stat Theor Methods 2005a;34:1957–1965. [CrossRef]
• [23] Sungur EA. Some observations on copula regression functions. Commun Stat Theor Methods 2005b;34:1967–1978. [CrossRef]
• [24] Rodríguez-Lallena JA, Úbeda-Flores MA. A new class of bivariate copulas. Stat Probab Lett 2004;66:315–325. [CrossRef]
• [25] Klement EP, Mesiar R. How non-symmetric can a copula be? Commen Math Univ Carolinae 2006;47:141148.
• [26] Grimaldi S, Serinaldi F. Asymmetric copula in multivariate flood frequency analysis. Adv. Water Resour 2006;29:1155–1167. [CrossRef]
• [27] Mesiar R, Najjari V. New families of symmetric/asymmetric copulas. Fuzzy Sets Syst 2014;252:99–110. [CrossRef]
• [28] Mazo G, Girard S, Forbes F. A class of multivariate copulas based on products of bivariate copulas. J Multivariate Anal 2015;140:363376. [CrossRef]
• [29] Khoudraji A. Contributions`al’ ́etude des copules et`alamod ́elisation des valeurs extrˆemes bivari ́ees. PhD thesis, Canada: Universit ́ede Laval, Qu ́ebec; 1995.
• [30] Liebscher E. Construction of asymmetric multivariate copulas. J Multivariate Anal 2008;99:2234–2250. [CrossRef]
• [31] Durante F. Construction of non-exchangeable bivariate distribution functions. Stat Papers 2009;50:383391. [CrossRef]
• [32] Quessy JF, Kortbi O. Minimum-distance statistics for the selection of an asymmetric copula in Khoudraji's class of models. Stat Sinica Preprint. doi: 10.5705/ss.202014.0082. [CrossRef]
• [33] Siburg KF, Stehling K, Stoimenov PA, Weiß GN. An order of asymmetry in copulas, and implications for risk management. Insur Math Econ 2016;68:241247. [CrossRef]
• [34] Bezak N, Rusjan S, Kramar Fijavž M, Mikoš M, Šraj M. Estimation of suspended sediment loads using copula functions. Water 2017;9:628. [CrossRef]
• [35] Zhang Y, Kim CW, Beer M, Dai H, Soares CG. Modeling multivariate ocean data using asymmetric copulas. Coastal Eng 2018;135:91111. [CrossRef]
• [36] Zhang Y, Gomes, AT, Beer M, Neumann I, Nackenhorst U, Kim CW. Modeling asymmetric dependences among multivariate soil data for the geotechnical analysis–The asymmetric copula approach. Soils Found 2019;59:19601979. [CrossRef]
• [37] Lin Y, Dong S, Tao S. Modelling long-term joint distribution of significant wave height and mean zero-crossing wave period using a copula mixture. Ocean Eng 2020;197:106856. [CrossRef]
• [38] Bai X, Jiang H, Li C, Huang L. Joint probability distribution of coastal winds and waves using a log-transformed kernel density estimation and mixed copula approach. Ocean Eng 2020;216:107937. [CrossRef]
• [39] Huang W, Dong S. Joint distribution of significant wave height and zero-up-crossing wave period using mixture copula method. Ocean Eng 2021; 219:108305. [CrossRef]
• [40] Wu S. Construction of asymmetric copulas and its application in two-dimensional reliability modeling. Eur J Oper Res 2014;238:476–485. [CrossRef]
• [41] Shih JH, Emura T. On the copula correlation ratio and its generalization. J Multivariate Anal 2021;182:104708. [CrossRef]
• [42] Dutang C, Charpentier A. Package ‘CASdatasets’ , 2020. http://cas.uqam.ca/pub/web/CASdatasets-manual.pdf
• [43] Frees EW, Wang P. Credibility using copulas. North Am Act J 2005;9:3148. [CrossRef]
• [44] Jung YS, Kim JM, Kim J. New approach of directional dependence in exchange markets using generalized FGM copula function. Commun Stat Simul Comput 2008;37:772788. [CrossRef]
• [45] Kim D, Kim JM. Analysis of directional dependence using asymmetric copula-based regression models. J Stat Comput Simul 2014;84:1990–2010. [CrossRef]