A comparative study on modeling of dependence between claim severity and frequency with Archimedean copulas
Abstract
In the estimation of aggregate loss, claim severity and claim frequency are generally assumed independent. Although, the independence assumption is quite basic, it may cause underestimates or overestimates in calculations. The dependence in the estimation of aggregate loss can be included with the copula-based models. The joint cumulative distribution and the joint probability density functions of mixed variables such as continuous claim severity and discrete claim frequency can be obtained using the bivariate copula functions and the mixed copula approach. In this study, aggregate loss is modeled using the bivariate Archimedean copula functions considering the dependency between claim components. In the application part, the aggregate loss is calculated using different Archimedean copula functions. Different copula functions and different parameters for each copula are used to analyze the effect of copula type and parameter. Furthermore, aggregate loss in the presence of dependence between claims are estimated.
References
- F. Lundberg, 1903, I. Approximerad framstallning af sannolikhetsfunktionen: II. Aterforsakring af kollektivrisker. Almqvist & Wiksell.
- E. W. Frees, E. A. Valdez, 1998, Understanding relationships using copulas. North American actuarial journal, 2(1), 1-25
- X. K. Song, P. X. K. Song, 2007, Correlated data analysis: modeling, analytics, and applications. Springer Science & Business Media.
- R. Kastenmeier, 2008, Joint regression analysis of insurance claims and claim sizes, Technische Universitat München, Mathematical Sciences, Diploma Thesis.
- P. X. K. Song, M. Li, Y. Yuan, 2009, Joint regression analysis of correlated data using Gaussian copulas, Biometrics, 65(1), 60-68.
- N. Kolev, D. Paiva, 2009, Copula-based regression models: A survey. Journal of statistical planning and inference, 139(11), 3847-3856.
- C. Czado, R. Kastenmeier, E. C. Brechmann, A. Min, 2012, A mixed copula model for insurance claims and claim sizes, Scandinavian Actuarial Journal, 4, 278-305.
- N. Kramer, E. C. Brechmann, D. Silvestrini, C. Czado, 2013, Total loss estimation using copula-based regression models. Insurance: Mathematics and Economics, 53(3), 829-839.
- S. Gschlöβl, C. Czado, 2007, Spatial modelling of claim frequency and claim size in non-life insurance. Scandinavian Actuarial Journal, 3, 202-225.
- J. Garrido, C. Genest, J. Schulz, 2016, Generalized linear models for dependent frequency and severity of insurance claims. Insurance: Mathematics and Economics, 70, 205-215.
- H. Arvidson, S. Francke, 2007, Dependence in non-life insurance, UUDM Project Report, 23.
- S. Anastasiadis, S. Chukova, 2012, Multivariate insurance models: an overview. Insurance: Mathematics and Economics, 51(1), 222-227.
- S. Eryilmaz, S., 2017, On compound sums under dependence. Insurance: Mathematics and Economics, 72, 228-234.
- A. Sklar, 1959, 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.
- R. B. Nelsen, 2006, An introduction to copulas. Springer Science & Business Media.
- Y. K. Tse, 2009, Nonlife Actuarial Models: Theory, Methods and Evaluation, Cambridge University Press.
- S. A. Klugman, H. H. Panjer, G. E. Willmot, 2008, Loss models: from data to decisions, John Wiley & Sons.
- N. Kramer, D. Silvestrini, M. N. Kraemer, 2013, Package ‘CopulaRegression’.
- B. Ripley, B. Venables, D. M. Bates, K. Hornik, A. Gebhardt, D. Firth, M.B. Ripley, 2013, Package ‘mass’. Cran R.
- U. Schepsmeier, J. Stoeber, E. C. Brechmann, B. Graeler, T. Nagler, T. Erhardt, 2012, VineCopula: Statistical inference of vine copulas. R package version 1.
Hasar tutarı ve sayısı arasındaki bağımlılığın Arşimet kopulalar ile modellenmesi üzerine karşılaştırmalı bir çalışma
Abstract
Toplam hasar tahmininde, hasar tutarı ile hasar sayısı genellikle bağımsız varsayılmaktadır. Bağımsızlık varsayımı oldukça temel olmasına rağmen, hesaplamalarda az veya fazla tahminlere neden olabilir. Toplam hasar tahminindeki bağımlılık, kopula-temelli modeller ile dikkate alınabilir. Sürekli hasar tutarı ve kesikli hasar sayısı gibi karma değişkenlerin ortak kümülatif dağılım ve ortak olasılık yoğunluk fonksiyonu iki değişkenli kopula fonksiyonları ve karma kopula yaklaşımı kullanılarak elde edilebilir. Bu çalışmada toplam hasar, hasar bileşenleri arasındaki bağımlılık dikkate alınarak, Arşimet kopula fonksiyonları yardımıyla modellenmiştir. Uygulama bölümünde, toplam hasar farklı Arşimet kopulalar kullanılarak tahmin edilmiştir. Kopula türü ve parametresinin etkisini analiz etmek amacıyla farklı kopula fonksiyonları ve her bir kopula için farklı parametreler kullanılmıştır.
References
- F. Lundberg, 1903, I. Approximerad framstallning af sannolikhetsfunktionen: II. Aterforsakring af kollektivrisker. Almqvist & Wiksell.
- E. W. Frees, E. A. Valdez, 1998, Understanding relationships using copulas. North American actuarial journal, 2(1), 1-25
- X. K. Song, P. X. K. Song, 2007, Correlated data analysis: modeling, analytics, and applications. Springer Science & Business Media.
- R. Kastenmeier, 2008, Joint regression analysis of insurance claims and claim sizes, Technische Universitat München, Mathematical Sciences, Diploma Thesis.
- P. X. K. Song, M. Li, Y. Yuan, 2009, Joint regression analysis of correlated data using Gaussian copulas, Biometrics, 65(1), 60-68.
- N. Kolev, D. Paiva, 2009, Copula-based regression models: A survey. Journal of statistical planning and inference, 139(11), 3847-3856.
- C. Czado, R. Kastenmeier, E. C. Brechmann, A. Min, 2012, A mixed copula model for insurance claims and claim sizes, Scandinavian Actuarial Journal, 4, 278-305.
- N. Kramer, E. C. Brechmann, D. Silvestrini, C. Czado, 2013, Total loss estimation using copula-based regression models. Insurance: Mathematics and Economics, 53(3), 829-839.
- S. Gschlöβl, C. Czado, 2007, Spatial modelling of claim frequency and claim size in non-life insurance. Scandinavian Actuarial Journal, 3, 202-225.
- J. Garrido, C. Genest, J. Schulz, 2016, Generalized linear models for dependent frequency and severity of insurance claims. Insurance: Mathematics and Economics, 70, 205-215.
- H. Arvidson, S. Francke, 2007, Dependence in non-life insurance, UUDM Project Report, 23.
- S. Anastasiadis, S. Chukova, 2012, Multivariate insurance models: an overview. Insurance: Mathematics and Economics, 51(1), 222-227.
- S. Eryilmaz, S., 2017, On compound sums under dependence. Insurance: Mathematics and Economics, 72, 228-234.
- A. Sklar, 1959, 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.
- R. B. Nelsen, 2006, An introduction to copulas. Springer Science & Business Media.
- Y. K. Tse, 2009, Nonlife Actuarial Models: Theory, Methods and Evaluation, Cambridge University Press.
- S. A. Klugman, H. H. Panjer, G. E. Willmot, 2008, Loss models: from data to decisions, John Wiley & Sons.
- N. Kramer, D. Silvestrini, M. N. Kraemer, 2013, Package ‘CopulaRegression’.
- B. Ripley, B. Venables, D. M. Bates, K. Hornik, A. Gebhardt, D. Firth, M.B. Ripley, 2013, Package ‘mass’. Cran R.
- U. Schepsmeier, J. Stoeber, E. C. Brechmann, B. Graeler, T. Nagler, T. Erhardt, 2012, VineCopula: Statistical inference of vine copulas. R package version 1.
Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Articles |
| Authors | |
| Publication Date | June 30, 2020 |
| Published in Issue | Year 2020 Volume: 13 Issue: 1 |