Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T20:58:14.676Z Has data issue: false hasContentIssue false
  • English
  • Français

A robust random coefficient regression representation of the chain-ladder method

Published online by Cambridge University Press:  09 June 2021

Ioannis Badounas ,
Apostolos Bozikas  and
Georgios Pitselis
Ioannis Badounas
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, Piraeus, Greece
Apostolos Bozikas
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, Piraeus, Greece
Georgios Pitselis*
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, Piraeus, Greece Department of Mathematics and Statistics, Concordia University, Montreal, Canada
*
*Corresponding author. E-mail: pitselis@unipi.gr
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well known that the presence of outliers can mis-estimate (underestimate or overestimate) the overall reserve in the chain-ladder method, when we consider a linear regression model, based on the assumption that the coefficients are fixed and identical from one observation to another. By relaxing the usual regression assumptions and applying a regression with randomly varying coefficients, we have a similar phenomenon, i.e., mis-estimation of the overall reserves. The lack of robustness of loss reserving regression with random coefficients on incremental payment estimators leads to the development of this paper, aiming to apply robust statistical procedures to the loss reserving estimation when regression coefficients are random. Numerical results of the proposed method are illustrated and compared with the results that were obtained by linear regression with fixed coefficients.

Keywords

Loss reservingrandom coefficient regression modelsrobust estimationchain-ladder methodG22
Type
Original Research Paper
Information
Annals of Actuarial Science , Volume 16 , Issue 1 , March 2022 , pp. 151 - 182
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Institute and Faculty of Actuaries

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Badounas, I. & Pitselis, G. (2020). Loss reserving estimation with correlated run-off triangles in a quantile longitudinal model. Risks, 8, 14.CrossRefGoogle Scholar
Bianco, A.M., Ben, M.G. & Yohai, V.J. (2005). Robust estimation for linear regression with asymmetric errors. The Canadian Journal of Statistics, 33(4), 511–528.CrossRefGoogle Scholar
Busse, M., Müller, U. & Dacorogna, M. (2010). Robust estimation of reserve risk. ASTIN Bulletin, 40(2), 453–489.Google Scholar
Christofides, S. (1997). Regression models based on log-incremental payments. Claims Reserving Manual, Institute of Actuaries, London, 9.Google Scholar
Chukhrova, N. & Johannssen, A. (2017). State space models and the Kalman-Filter in stochastic claims reserving: Forecasting, filtering and smoothing. Risks, 5, 30.CrossRefGoogle Scholar
De Alba, E. (2002). Bayesian estimation of outstanding claim reserves. North American Actuarial Journal, 6(4), 1–20.CrossRefGoogle Scholar
Dornheim, H. & Brazauskas, V. (2011). Robust-efficient credibility models with heavy-tailed claims: A mixed linear models perspective. Insurance: Mathematics and Economics, 48, 72–84.Google Scholar
Gervini, D., & Yohai, V.J. (2002). A class of robust and fully efficient regression estimators. The Annals of Statistics, 30(2), 583–616.CrossRefGoogle Scholar
Greene, W.H. (2012). Econometric Analysis, 7th ed. International ed. Pearson Education Limited, London.Google Scholar
Griffiths, W.E. (1972). Estimation of actual response coefficients in the Hildreth-Houck random coefficient model. Journal of the American Statistical Association, 67(339), 633–635.CrossRefGoogle Scholar
Hampel, F.R., Ronchetti, E.M, Rousseeuw, P.J. & Stahel, W.A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley.Google Scholar
Hildreth, C. & Houck, J.P. (1968). Some estimators for a linear model with random coefficients. Journal of the American Statistical Association, 63(322), 584–595.Google Scholar
Hsiao, C. (2003). Analysis of Panel Data, 2nd ed. Cambridge University Press, New York.CrossRefGoogle Scholar
Huber, P.J. (1981). Robust Statistics. Wiley.CrossRefGoogle Scholar
Huber, P.J & Dutter, R. (1974). Numerical solutions of robust regression. In Bruckman, G., Ferschl, F. & Schmetterer, L. (Eds.), COMPSTAT: Proceedings in Computational Statistics (pp. 165–172). Physica-Verlag.Google Scholar
Hubert, M., Verdonck, T. & Yorulmaz, O. (2017). Fast robust SUR with economical and actuarial applications. Statistical Analysis and Data Mining, 10(2), 77–88.CrossRefGoogle Scholar
Jurečková, J., Picek, J. & Schindler, M. (2019). Robust Statistical Methods with R. CRC Press.CrossRefGoogle Scholar
De Jong, P. & Zehnwirth, B. (1983). Claims reserving, state-space models and the Kalman filter. Journal of the Institute of Actuaries, 110(1), 157–181.CrossRefGoogle Scholar
Koller, M. & Stahel, W.A. (2011). Sharpening Wald-type inference in robust regression for small samples. Computational Statistics & Data Analysis, 55(8), 2504–2515.CrossRefGoogle Scholar
Kremer, E. (1997). Robust lagfactors. Blätter der DGVFM, 23(2), 137–145.CrossRefGoogle Scholar
Maronna, R.A., Martin, D.R. & Yohai, V.J. (2006). Robust Statistics: Theory and Methods. Wiley.CrossRefGoogle Scholar
Peremans, K., Van Aelst, S. & Verdonck, T. (2018). A robust general multivariate chain ladder method. Risks, 6(4), 1–18.CrossRefGoogle Scholar
Maechler, M., Rousseeuw, P., Croux, C., Todorov, V., Ruckstuhl, A., Salibian-Barrera, M., Verbeke, T., Koller, M., Conceicao, E.L.T. & di Palma, M.A. (2020). Robustbase: Basic robust statistics R package version 0.93-6. Available online at the address http://CRAN.R-project.org/package=robustbase Google Scholar
Pitselis, G. (2014). Robust eligible own funds and value at risk under solvency II system. Communication in Statistics – Simulation and Computation, 43(1), 161–182.CrossRefGoogle Scholar
Pitselis, G., Grigoriadou, V. & Badounas, I. (2015). Robust loss reserving in a log-linear model. Insurance: Mathematics and Economics, 64, 14–27.Google Scholar
R Core Team, R (2020). A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Available online at the address https://www.R-project.org/ Google Scholar
Rousseeuw, P.J. & Leroy, A.M. (1987) Robust Regression and Outlier Detection. Wiley.CrossRefGoogle Scholar
Swamy, P.A.V.B. (1971). Statistical Inference in Random Coefficient Regression Models. Springer-Verlag, New York.CrossRefGoogle Scholar
Taylor, G.C. & Ashe, F.R. (1983). Second moments of estimates of outstanding claims. Journal of Econometrics, 23(1), 37–61.CrossRefGoogle Scholar
Venables, W.N. & Ripley, B.D. (2002). Modern Applied Statistics with S, 4th ed. Springer, New York.CrossRefGoogle Scholar
Verdonck, T., Van Wouwe, M. & Dhaene, J. (2009). A robustification of the Chain-Ladder method. North American Actuarial Journal, 13(2), 280–298.CrossRefGoogle Scholar
Verdonck, T. & Debruyne, M. (2011). The influence of individual claims on the Chain-Ladder estimates: Analysis and diagnostic tool. Insurance: Mathematics and Economics, 48(1), 85–98.Google Scholar
Verrall, R.J. (1989). A state space representation of the Chain Ladder linear model. Journal of the Institute of Actuaries, 3, 589–609.CrossRefGoogle Scholar
Verrall, R.J. (1994). A method for modelling varying run-off evolutions in claims reserving. Astin Bulletin, 24(2), 325–332.CrossRefGoogle Scholar
Wang, Y.G., Lin, X. & Zhu, M. (2005). Robust estimating functions and bias correction for longitudinal data analysis. Biometrics, 61(3), 684–691.CrossRefGoogle ScholarPubMed
Wang, J., Zamar, R., Marazzi, A., Yohai, V., Salibian-Barrera, M., Maronna, R., Zivot, E., Rocke, D., Martin, D., Maechler, M. & Konis, K. (2020). Robust: Port of the S+ “Robust Library”. R package version 0.5-0.0. Available online at the address https://CRAN.R-project.org/package=robust Google Scholar
Wüthrich, M. (2007). Using a Bayesian approach for claims reserving. Variance, 1(2), 292–301.Google Scholar
Yohai, V.J. (1987). High breakdown-point and high efficiency robust estimates for regression. The Annals of Statistics, 15, 642–656.CrossRefGoogle Scholar
Yohai, V.J. & Zamar, R.H. (1997). Optimal locally robust M-estimates of regression. Journal of Statistical Planning and Inference, 64(2), 309–323.CrossRefGoogle Scholar
Zhang, Y. (2010). A general multivariate Chain Ladder model. Insurance: Mathematics and Economics, 46(3), 588–599.Google Scholar