Book contents
- Frontmatter
- Contents
- Contributor List
- Acknowledgments
- 1 Predictive Modeling in Actuarial Science
- II Predictive Modeling Foundations
- II Predictive Modeling Methods
- 7 Longitudinal and Panel Data Models
- 8 Linear Mixed Models
- 9 Credibility and Regression Modeling
- 10 Fat-Tailed Regression Models
- 11 Spatial Modeling
- 12 Unsupervised Learning
- III Bayesian and Mixed Modeling
- IV Longitudinal Modeling
- Index
- References
8 - Linear Mixed Models
from II - Predictive Modeling Methods
Published online by Cambridge University Press: 05 August 2014
Book contents
- Frontmatter
- Contents
- Contributor List
- Acknowledgments
- 1 Predictive Modeling in Actuarial Science
- II Predictive Modeling Foundations
- II Predictive Modeling Methods
- 7 Longitudinal and Panel Data Models
- 8 Linear Mixed Models
- 9 Credibility and Regression Modeling
- 10 Fat-Tailed Regression Models
- 11 Spatial Modeling
- 12 Unsupervised Learning
- III Bayesian and Mixed Modeling
- IV Longitudinal Modeling
- Index
- References
Summary
Chapter Preview. We give a general discussion of linear mixed models and continue by illustrating specific actuarial applications of this type of model. Technical details on linear mixed models follow: model assumptions, specifications, estimation techniques, and methods of inference. We include three worked-out examples with the R lme4 package and use ggplot2 for the graphs. Full code is available on the book's website.
Mixed Models in Actuarial Science
8.1.1 What Are Linear Mixed Models?
A First Example of a Linear Mixed Model. As explained in Chapter 7, a panel dataset follows a group of subjects (e.g., policyholders in an insurance portfolio) over time. We therefore denote variables (e.g., yit, xit) in a panel dataset with double subscripts, indicating the subject (say, i) and the time period (say, t). As motivated in Section 1.2 of Chapter 7, the analysis of panel data has several advantages. Panel data allow one to study the effect of certain covariates on the response of interest (as in usual regression models for cross–sectional data), while accounting appropriately for the dynamics in these relations. For actuarial ratemaking the availability of panel data is of particular interest in a posteriori ratemaking. An a posteriori tariff predicts the current year loss for a particular policyholder, using (among other factors) the dependence between the current year's loss and losses reported by this policyholder in previous years.
Information
- Type
- Chapter
- Information
- Predictive Modeling Applications in Actuarial Science , pp. 182 - 216Publisher: Cambridge University PressPrint publication year: 2014
References
and (2007). Actuarial statistics with generalized linear mixed models. Insurance: Mathematics and Economics 40(1), 58–76.
and (2008). Issues in claims reserving and credibility: A semiparametric approach with mixed models. Journal of Risk and Insurance 75(3), 643–676.CrossRefGoogle Scholar
, , and (2010). A multilevel analysis of intercompany claim counts. ASTIN Bulletin 40(1), 151–177.CrossRefGoogle Scholar
and (2012). Statistical aspects of apriori and a posteriori risk classification in insurance. Advances in Statistical Analysis 96(2), 187–224.CrossRefGoogle Scholar
and (2005). A Course in Credibility Theory and Its Applications. Springer Verlag, Berlin.Google Scholar
, , and (1996). Practical Actuarial Credibility Models. Institute of Actuarial Science and Econometrics, University of Amsterdam.Google Scholar
(2004). Longitudinal and Panel Data. Analysis and Applications in the Social Sciences. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
, , and (1999). A longitudinal data analysis interpretation of credibility models. Insurance: Mathematics and Economics 24(3), 229–247.Google Scholar
, , and (2001). Case studies using panel data models. North American Actuarial Journal 5(4), 24–42.CrossRefGoogle Scholar
(2006). Multilevel (hierarchical) modeling: What it can and cannot do. Technometrics 48(3), 432–135.CrossRefGoogle Scholar
and (2007). Applied Regression and Multilevel (Hierarchical) Models. Cambridge University Press, Cambridge.Google Scholar
(1975). In Credibility: Theory and Applications, Credibility for regression models with application to trend, pp. 129–163. Academic Press, New York.Google Scholar
and (1984). Approximations for standard errors of estimators of fixed and random effects in mixed linear models. Journal of the American Statistical Association 79, 853–862.Google Scholar
(1992). Bayesian Statistics in Actuarial Science with Emphasis on Credibility. Kluwer, Boston.CrossRefGoogle Scholar
and (1982). Random-effects models for longitudinal data. Biometrics 38(4), 963–974.CrossRefGoogle ScholarPubMed
, , and (1996). Bayesian methods in actuarial science. The Statistician 45(4), 503–515.CrossRefGoogle Scholar
and (2001). Generalized, Linear and Mixed Models. Wiley Series in Probability and Statistics, Wiley, New York.Google Scholar
(1991). That BLUP is a good thing: The estimation of random effects. Statistical Science 6, 15–51.Google Scholar
, , and (2003). Semiparametric Regression. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
(1996). An introduction to Markov Chain Monte Carlo methods and their actuarial applications. Proceedings of the Casualty Actuarial Society Forum LXXXIII, 114–165.Google Scholar
and (2000). Linear Mixed Models for Longitudinal Data. Springer Series in Statistics, New York.Google Scholar
(2006). Generalized Additive Models: An Introduction with R. Chapman & Hall, CRC Texts in Statistical Science.Google Scholar
and (2012). Predicting multivariate insurance loss payments under the Bayesian copula framework. Journal of Risk and Insurance.
, , and (2012). A Bayesian nonlinear model for forecasting insurance loss payments. Journal of the Royal Statistical Society, Series A 175, 637–656.CrossRefGoogle Scholar
Accessibility standard: Unknown
Why this information is here
This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.Accessibility Information
Accessibility compliance for the PDF of this chapter is currently unknown
and may be updated in the future.
- 3
- Cited by