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Calculation of LTC Premiums Based on Direct Estimates of Transition Probabilities

Published online by Cambridge University Press:  17 April 2015

Florian Helms ,
Claudia Czado  and
Susanne Gschlößl
Florian Helms
Affiliation:
Technische Universität München, Zentrum Mathematik, Email: [email protected]
Claudia Czado
Affiliation:
Technische Universität München, Zentrum Mathematik, Email: [email protected]
Susanne Gschlößl
Affiliation:
Technische Universität München, Zentrum Mathematik, Email: [email protected]
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Abstract

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In this paper we model the life-history of LTC-patients using a Markovian multi-state model in order to calculate premiums for a given LTC-plan. Instead of estimating the transition intensities in this model we use the approach suggested by Andersen et al. (2003) for a direct estimation of the transition probabilities. Based on the Aalen-Johansen estimator, an almost unbiased estimator for the transition matrix of a Markovian multi-state model, we calculate so-called pseudo-values, known from Jackknife methods. Further, we assume that the relationship between these pseudo-values and the covariates of our data are given by a GLM with the logit as link-function. Since the GLMs do not allow for correlation between successive observations we use instead the “Generalized Estimating Equations” (GEEs) to estimate the parameters of our regression model. The approach is illustrated using a representative sample from a German LTC portfolio.

Keywords

Markovian Multi-State ModelTransition ProbabilitiesAalen-Johansen EstimatorPseudo-ValuesGLMGEELTCPremium
Type
Workshop
Information
ASTIN Bulletin: The Journal of the IAA , Volume 35 , Issue 2 , November 2005 , pp. 455 - 469
Copyright
Copyright © ASTIN Bulletin 2005

References

Aalen, O.O. and Johansen, S. (1978) An Empirical Transition Matrix for Non-homogeneous Markov Chains based on Censored Observations. Scand J Statist., 5, 141–150.Google Scholar
Andersen, P.K., Borgan, Ø., Gill, R.D. and Keiding, N. (1993) Statistical Models Based on Counting Processes. New York: Springer-Verlag.CrossRefGoogle Scholar
Andersen, P.K., Klein, J.P. and Rosthøj, S. (2003) Generalised Linear Models for correlated Pseudo-Observations, with Applications to Multi-State Models. Biometrika 90, 1, 15–27.CrossRefGoogle Scholar
COE (2003) Recent demographic developments in europe 2002. www.coe.int.Google Scholar
Collett, D. (2002) Modelling Binary Data 2nd ed. London: Chapman & Hall.CrossRefGoogle Scholar
Czado, C. and Rudolph, F. (2002) Application of Survival Analysis Methods to Long Term Care Insurance. Insurance: Mathematics and Economics, 31, 395–413.Google Scholar
Efron, B. and Tibshirani, R.J. (1993) An Introduction to the Bootstrap. New York: Chapman & Hall.CrossRefGoogle Scholar
Fahrmeir, L. and Tutz, G. (1994) Multivariate Statistical Modelling Based On Generalized Linear Models. New York: Springer.CrossRefGoogle Scholar
Gamerman, D. (1990) Stochastic simulation for Bayesian inference. London: Champman & Hall.Google Scholar
Gilks, W.R., Richardson, W.S., and Spiegelhalter, D. (1996) Markov Chain Monte Carlo in Practice. London: Chapman & Hall.Google Scholar
Haberman, S. and Pitacco, E. (1999) Actuarial Models for Disability Insurance. London: Chapman & Hall.Google Scholar
Hardin, J.W. and Hilbe, J.M. (2003) Generalized Estimating Equations, Chapman & Hall / CRC, Boca Raton, FL.Google Scholar
Helms, F. (2003) Estimating LTC-Premiums using GEEs for Peudo-Values. Technische Universität München Diplomarbeit.Google Scholar
Kaas, R.E. and Raftery, A.E. (1995) Bayes factors. Journal of the American Statistical Association, 90, 773–795.CrossRefGoogle Scholar
Kaplan, E.L. and Meier, P. (1958) Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53 (282), 457–481.CrossRefGoogle Scholar
Liang, K.-Y. and Zeger, S.L. (1986) Longitudinal Data Analysis using Generalized Linear Models. Biometrika, 73(1), 13–221.CrossRefGoogle Scholar
McCullagh, P. and Nelder, J.A. (1989) Generalized Linear Models 2nd ed. London: Chapman and Hall.CrossRefGoogle Scholar
Rudolph, F. (2000) Anwendungen der Überlebenzeitanalyse in der Pflegeversicherung. Technische Universität München Diplomarbeit.Google Scholar
Spiegelhalter, D., Best, N., Carlin, B. and van der Linde, A. (2002) Bayesian measures of model complexity and fit. J. Roy. Stat. Soc. B., 64, 583–639.CrossRefGoogle Scholar
Wedderburn, R.W.M. (1974) Quasi-Likelihood Functions, Generalized Linear Models, and the Gauss-Newton Method. Biometrika 61(3), 439–447.Google Scholar
Zeger, S.L. and Liang, K.-Y. (1986) Longitudinal Data Analysis for Discrete and Continuous Outcomes. Biometrics, 42(1), 121–130.CrossRefGoogle ScholarPubMed