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Genelleştirilmiş Doğrusal Modellerin İki Değişkenli Gauss Kopula'ya Dahil Edilmesi ve Bir Uygulama

Year 2022, Issue: 5, 1 - 9, 30.06.2022
https://doi.org/10.52693/jsas.1039360

Abstract

Hayat dışı sigorta matematiğinde, son yıllarda hasar değişkenleri arasında bağımlılık varlığında analizler ve prim veya karşılık hesaplamaları yapılmaktadır. Ve böylece, iddianın ciddiyeti ve sıklığı arasındaki bağımlılık varsayımından kaynaklanan toplam kaybın gereğinden fazla veya eksik tahmin edilmesi engellenir. Bağımlılık modellemesi için sıklıkla kullanılan Gauss kopula işlevi, "kopula regresyonu" adı verilen karma kopula tabanlı bir regresyon modeli elde etmek için marjinal genelleştirilmiş doğrusal modellere entegre edilmiştir. Bu çalışmada, talep şiddeti ve frekansı için sırasıyla iki değişkenli Gauss kopula, Gamma ve Poisson marjinal genelleştirilmiş doğrusal modeller kullanılarak bir kopula regresyon modeli oluşturulmuştur. R paketi “CopulaRegression” kullanılarak, iddianın ciddiyeti ve sıklığı arasında bir bağımlılığın olduğu, simüle edilmiş bir veri ile bir uygulama gerçekleştirilir. İstemler arasındaki bağımlılığı modellemenin önemi, bağımsız ve bağımlı modellerin karşılaştırılmasıyla araştırılmış ve uygulama sonuçları, bağımlılığın dikkate alındığı kopula regresyon modelinin, bağımsız marjinal genelleştirilmiş doğrusal modellere kıyasla daha düşük nispi ortalama kare hatalarına sahip olduğunu göstermektedir.

References

  • [1] C. Czado, R. Kastenmeier, E. C. Brechmann and A. Min, “A mixed copula model for insurance claims and claim sizes”, Scand. Actuar. J., vol. 4, pp. 278-305, 2012.
  • [2] Y. K. Tse. “Nonlife actuarial models: theory, methods and evaluation”, Cambridge University Press, 2009.
  • [3] E. Ohlsson, B. Johansson, “Non-life insurance pricing with generalized linear models”, Springer, 174, 2010.
  • [4] M. David, “Automobile insurance pricing with generalized linear models”, Proceedings in GV- The 3rd Global Vitual Conference, 6-10 April, 2015.
  • [5] E. W. Frees and E. A. Valdez, “Understanding relationships using copulas”, N. Am. Actuar. J., vol. 2, pp. 1-25, 1998.
  • [6] P. X. K. Song, “Correlated data analysis: modeling, analytics, and applications”, Springer Science & Business Media, 2007.
  • [7] P. X. K. Song, M. Li and Y. Yuan, “Joint regression analysis of correlated data using Gaussian copulas” , Biometrics, vol. 65(1), pp. 60-68, 2009.
  • [8] R. Kastenmeirer, “Joint regression analysis of insurance claims and claim sizes”, Diploma Thesis, Technische Universitat München, Mathematical Sciences, 2008. [9] N. Kolev and D. Pavia, “Copula-based regression models: A survey”. J Stat Plan Inference, vol. 139(11), pp. 3847-3856, 2009.
  • [10] A. R. De Leon and B. Wu, “Copula‐based regression models for a bivariate mixed discrete and continuous outcome”, Stat Med, vol. 30(2), pp. 175-185, 2011.
  • [11] N. Krämer, E. C. Brechmann, D. Silvestrini and C. Czado, “Total loss estimation using copula-based regression models”, Insur Math Econ, vol. 53(3), pp. 829-839, 2013. [12] S. Gschlöβl and C. Czado, “Spatial modelling of claim frequency and claim size in non-life insurance” Scand, vol. 3, pp. 202-225, 2007.
  • [13] J. Garrido, C. Genest and J. Schulz, “Generalized linear models for dependent frequency and severity of insurance claims”, Insur Math Econ, vol. 70, pp. 205-215, 2016. [14] P. Shi, “Insurance ratemaking using a copula-based multivariate Tweedie model”, Scand. Actuar. J., vol. 3, pp. 198-215, 2016.
  • [15] A. T. Payandeh Najafabadi, M. Qazvini, “A GLM approach to estimating copula models”, Comm. Statist. Simulation Comput., vol. 44 (6), pp. 1641-1656, 2015. [16] N. Krämer, D. Silvestrini and M. N. Krämer, Package ‘CopulaRegression’, 2013.
  • [17] A. Sklar, “Fonctions de répartition à n dimensions et leurs marges”, Publications de l’Institut de Statistique de L’Université de Paris, vol. 8, pp. 229-231, 1959. [18] R. B. Nelsen, “An introduction to copulas”, Springer Science & Business Media, 2006.
  • [19] D. Brigo, A. Pallavicini and R. Torresetti, “Credit Models and The Crisis: A Journey Into Cdos, Copulas”, Correlations And Dynamic Models, John Wiley & Sons, 2010.
  • [20] P. McCullagh and J. A. Nelder, “Generalized Linear Models”, CRC press, 37, 1989.
  • [21] B. Ripley, B. Venables, D. M. Bates, K. Hornik, A. Gebhardt, D. Firth and M. B. Ripley, Package ‘mass’, Cran R, 2013.
  • [22] U. Schepsmeier, J. Stoeber, E. C. Brechmann, B. Graeler, T. Nagler and T. Erhardt, “VineCopula: Statistical Inference of Vine Copulas”, R package version 1, 2012.
  • [23]Ö. K. Erdemir and M. Sucu, “A comparative study on modeling of dependency between claim severity and frequency”, J. Stat.: Stat and Actuar. Sci., vol. 13(1), pp. 18-29, 2020.

The Incorporation of Generalized Linear Models into Bivariate Gaussian Copula and An Application

Year 2022, Issue: 5, 1 - 9, 30.06.2022
https://doi.org/10.52693/jsas.1039360

Abstract

In non-life insurance mathematics, analyses and premium or reserve calculations are carried out in the presence of dependency between the claim variables in recent years. And, thus over- or underestimation of aggregate loss caused by the assumption of dependency between the claim severity and frequency are prevented. The Gaussian copula function, which is frequently used for dependency modeling, is integrated into the marginal generalized linear models to obtain a mixed copula-based regression model called "copula regression". In this study, a copula regression model is created using a bivariate Gaussian copula, Gamma and Poisson marginal generalized linear models for claim severity and frequency, respectively. An application is performed with a simulated data where there is a dependence between the claim severity and frequency using the R package “CopulaRegression”. The importance of the modeling of dependency between claims is investigated by the comparison of the independent and dependent models and the results of application show that the copula regression model in which dependency is considered has lower relative mean square errors compared the independent marginal generalized linear models.

References

  • [1] C. Czado, R. Kastenmeier, E. C. Brechmann and A. Min, “A mixed copula model for insurance claims and claim sizes”, Scand. Actuar. J., vol. 4, pp. 278-305, 2012.
  • [2] Y. K. Tse. “Nonlife actuarial models: theory, methods and evaluation”, Cambridge University Press, 2009.
  • [3] E. Ohlsson, B. Johansson, “Non-life insurance pricing with generalized linear models”, Springer, 174, 2010.
  • [4] M. David, “Automobile insurance pricing with generalized linear models”, Proceedings in GV- The 3rd Global Vitual Conference, 6-10 April, 2015.
  • [5] E. W. Frees and E. A. Valdez, “Understanding relationships using copulas”, N. Am. Actuar. J., vol. 2, pp. 1-25, 1998.
  • [6] P. X. K. Song, “Correlated data analysis: modeling, analytics, and applications”, Springer Science & Business Media, 2007.
  • [7] P. X. K. Song, M. Li and Y. Yuan, “Joint regression analysis of correlated data using Gaussian copulas” , Biometrics, vol. 65(1), pp. 60-68, 2009.
  • [8] R. Kastenmeirer, “Joint regression analysis of insurance claims and claim sizes”, Diploma Thesis, Technische Universitat München, Mathematical Sciences, 2008. [9] N. Kolev and D. Pavia, “Copula-based regression models: A survey”. J Stat Plan Inference, vol. 139(11), pp. 3847-3856, 2009.
  • [10] A. R. De Leon and B. Wu, “Copula‐based regression models for a bivariate mixed discrete and continuous outcome”, Stat Med, vol. 30(2), pp. 175-185, 2011.
  • [11] N. Krämer, E. C. Brechmann, D. Silvestrini and C. Czado, “Total loss estimation using copula-based regression models”, Insur Math Econ, vol. 53(3), pp. 829-839, 2013. [12] S. Gschlöβl and C. Czado, “Spatial modelling of claim frequency and claim size in non-life insurance” Scand, vol. 3, pp. 202-225, 2007.
  • [13] J. Garrido, C. Genest and J. Schulz, “Generalized linear models for dependent frequency and severity of insurance claims”, Insur Math Econ, vol. 70, pp. 205-215, 2016. [14] P. Shi, “Insurance ratemaking using a copula-based multivariate Tweedie model”, Scand. Actuar. J., vol. 3, pp. 198-215, 2016.
  • [15] A. T. Payandeh Najafabadi, M. Qazvini, “A GLM approach to estimating copula models”, Comm. Statist. Simulation Comput., vol. 44 (6), pp. 1641-1656, 2015. [16] N. Krämer, D. Silvestrini and M. N. Krämer, Package ‘CopulaRegression’, 2013.
  • [17] A. Sklar, “Fonctions de répartition à n dimensions et leurs marges”, Publications de l’Institut de Statistique de L’Université de Paris, vol. 8, pp. 229-231, 1959. [18] R. B. Nelsen, “An introduction to copulas”, Springer Science & Business Media, 2006.
  • [19] D. Brigo, A. Pallavicini and R. Torresetti, “Credit Models and The Crisis: A Journey Into Cdos, Copulas”, Correlations And Dynamic Models, John Wiley & Sons, 2010.
  • [20] P. McCullagh and J. A. Nelder, “Generalized Linear Models”, CRC press, 37, 1989.
  • [21] B. Ripley, B. Venables, D. M. Bates, K. Hornik, A. Gebhardt, D. Firth and M. B. Ripley, Package ‘mass’, Cran R, 2013.
  • [22] U. Schepsmeier, J. Stoeber, E. C. Brechmann, B. Graeler, T. Nagler and T. Erhardt, “VineCopula: Statistical Inference of Vine Copulas”, R package version 1, 2012.
  • [23]Ö. K. Erdemir and M. Sucu, “A comparative study on modeling of dependency between claim severity and frequency”, J. Stat.: Stat and Actuar. Sci., vol. 13(1), pp. 18-29, 2020.
There are 18 citations in total.

Details

Primary Language English
Subjects Statistics, Finance
Journal Section Research Articles
Authors

Övgücan Karadağ Erdemir 0000-0002-4725-3588

Meral Sucu 0000-0002-7991-1792

Early Pub Date June 30, 2022
Publication Date June 30, 2022
Published in Issue Year 2022 Issue: 5

Cite

IEEE Ö. K. Erdemir and M. Sucu, “The Incorporation of Generalized Linear Models into Bivariate Gaussian Copula and An Application”, JSAS, no. 5, pp. 1–9, June 2022, doi: 10.52693/jsas.1039360.