A longitudinal data analysis interpretation of credibility models

Abstract In this paper, we develop links between credibility theory in actuarial science and longitudinal data models in statistics. Our primary contribution to actuarial science is to demonstrate that many additive credibility models can be expressed as special cases of the longitudinal data model. We, thereby, unify the many existing credibility models with this framework. In addition, a longitudinal data interpretation suggests additional models and techniques that actuaries can use in credibility ratemaking. We also apply standard statistical software, which has been developed to analyze longitudinal data models, to the private passenger automobile data of Hachemeister [Hachemeister, C.A., 1975. Credibility for regression models with applications to trend. In: Kahn, P.M. (Ed.), Credibility: Theory and Applications. Academic Press, New York, pp. 129–163].

[1]  Stuart A. Klugman,et al.  Loss Models: From Data to Decisions , 1998 .

[2]  Ragnar Norberg,et al.  Hierarchical credibility: analysis of a random effect linear model with nested classification , 1986 .

[3]  Stuart A. Klugman,et al.  Credibility Models with Time-Varying Trend Components , 1991 .

[4]  Richard Verrall,et al.  Credibility Theory and Generalized Linear Models , 1997, ASTIN Bulletin.

[5]  Rob Kaas,et al.  Practical actuarial credibility models , 1996 .

[6]  R. Potthoff,et al.  A generalized multivariate analysis of variance model useful especially for growth curve problems , 1964 .

[7]  D. Hand,et al.  Practical Longitudinal Data Analysis , 1996 .

[8]  Stephen E. Fienberg,et al.  Credibility: Theory and Applications. , 1975 .

[9]  G. Robinson.  That BLUP is a Good Thing: The Estimation of Random Effects , 1991 .

[10]  D. Harville.  Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems , 1977 .

[11]  F. De Vylder.  PRACTICAL CREDIBILITY THEORY WITH EMPHASIS ON OPTIMAL PARAMETER ESTIMATION , 1981 .

[12]  P. Swamy.  Efficient Inference in a Random Coefficient Regression Model , 1970 .

[13]  Stuart A. Klugman.  Bayesian Statistics in Actuarial Science , 1992 .

[14]  A. Goldberger.  Best Linear Unbiased Prediction in the Generalized Linear Regression Model , 1962 .

[15]  Russ Wolfinger,et al.  Computing Gaussian Likelihoods and Their Derivatives for General Linear Mixed Models , 1994, SIAM J. Sci. Comput..

[16]  Ragnar Norberg,et al.  Empirical Bayes credibility , 1980 .

[17]  S. R. Searle,et al.  Minimum Variance Quadratic Unbiased Estimation (MIVQUE) of Variance Components , 1978 .

[18]  Patrick Sevestre,et al.  The Econometrics of Panel Data , 1993 .

[19]  D. Harville.  Bayesian inference for variance components using only error contrasts , 1974 .

[20]  S. R. Searle,et al.  A Comparison of Variance Component Estimators , 1976 .

[21]  C. R. Henderson.  SIRE EVALUATION AND GENETIC TRENDS , 1973 .

[22]  C. R. Rao,et al.  Estimation of Variance and Covariance Components in Linear Models , 1972 .

[23]  F. Graybill,et al.  Matrices with Applications in Statistics. , 1984 .

[24]  Ragnar Norberg,et al.  On optimal parameter estimation in credibility , 1982 .

[25]  Hans Bühlmann,et al.  Experience rating and credibility , 1967, ASTIN Bulletin.

[26]  Cheng Hsiao,et al.  Analysis of Panel Data , 1987 .

[27]  Marc Goovaerts,et al.  Exact Credibility for Weighted Observations , 1997, ASTIN Bulletin.

[28]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[29]  H. D. Patterson,et al.  Recovery of inter-block information when block sizes are unequal , 1971 .

[30]  P. Diggle,et al.  Analysis of Longitudinal Data , 2003 .

[31]  D. R. Dannenburg.  Some results on the estimation of the credibility factor in the classical Bühlmann model , 1994 .

[32]  Alois Gisler,et al.  Credibility in the Regression case Revisited (A Late Tribute to Charles A. Hachemeister) , 1997 .

[33]  B. Baltagi,et al.  Econometric Analysis of Panel Data , 2020, Springer Texts in Business and Economics.

[34]  David A. Harville,et al.  Decomposition of Prediction Error , 1985 .

[35]  John A. Nelder,et al.  Generalized linear models. 2nd ed. , 1993 .

[36]  S. R. Searle,et al.  Restricted Maximum Likelihood (REML) Estimation of Variance Components in the Mixed Model , 1976 .

[37]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[38]  J. Nelder,et al.  Hierarchical Generalized Linear Models , 1996 .