Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-23T13:32:28.326Z Has data issue: false hasContentIssue false
  • English
  • Français

POISSON MODELS WITH DYNAMIC RANDOM EFFECTS AND NONNEGATIVE CREDIBILITIES PER PERIOD

Published online by Cambridge University Press:  13 February 2020

Jean Pinquet
Jean Pinquet*
Affiliation:
Department of Mathematics and Economics, University of Paris Nanterre, 200, avenue de la République, 92001Nanterre Cedex, France
*
E-mails: pinquet@parisnanterre.fr; jpinquet@free.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper provides a toolbox for the credibility analysis of frequency risks, with allowance for the seniority of claims and of risk exposure. We use Poisson models with dynamic and second-order stationary random effects that ensure nonnegative credibilities per period. We specify classes of autocovariance functions that are compatible with positive random effects and that entail nonnegative credibilities regardless of the risk exposure. Random effects with nonnegative generalized partial autocorrelations are shown to imply nonnegative credibilities. This holds for ARFIMA(0, d, 0) models. The AR(p) time series that ensure nonnegative credibilities are specified from their precision matrices. The compatibility of these semiparametric models with log-Gaussian random effects is verified. Gaussian sequences with ARFIMA(0, d, 0) specifications, which are then exponentiated entrywise, provide positive random effects that also imply nonnegative credibilities. Dynamic random effects applied to Poisson distributions are retained as products of two uncorrelated and positive components: the first is time-invariant, whereas the autocovariance function of the second vanishes at infinity and ensures nonnegative credibilities. The limit credibility is related to the three levels for the length of the memory in the random effects. The limit credibility is less than one in the short memory case, and a formula is provided.

Keywords

Credibilitylinear filteringprecision matricesPerron–Frobeniusgeneralized partial autocorrelation coefficientsLevinson–Durbin recursionhereditarityWold decompositionspectral measureergodicitygeneralized method of momentslimit credibilityC14C18C23
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 2 , May 2020 , pp. 585 - 618
Copyright
© Astin Bulletin 2020

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

Angers, J.-F., Desjardins, D., Dionne, G. and Guertin, F. (2018) Modelling and estimating individual and firm effects with count panel data. ASTIN Bulletin, 48(3), 1049–1078.CrossRefGoogle Scholar
Ash, R.B. and Gardner, M.F. (1975) Topics in Stochastic Processes, Vol. 27, New York: Academic Press.Google Scholar
Berman, A. and Plemmons, R.J. (1994) Nonnegative Matrices in the Mathematical Sciences, Philadelphia: SIAM.CrossRefGoogle Scholar
Bolancé, C., Guillén, M. and Pinquet, J. (2003) Time-varying credibility for frequency risk models: Estimation and tests for autoregressive specifications on the random effects. Insurance: Mathematics and Economics, 33(2), 273–282.Google Scholar
Bühlmann, H. (1967) Experience rating and credibility. ASTIN Bulletin: The Journal of the IAA, 4(3), 199–207.CrossRefGoogle Scholar
Bühlmann, H. and Gisler, A. (2006) A Course in Credibility Theory and Its Applications, Heidelberg: Springer.Google Scholar
Cramér, H. and Leadbetter, M.R. (1967) Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications, New York: Courier Corporation.Google Scholar
Denuit, M., Guillen, M. and Trufin, J. (2019) Multivariate credibility modelling for usage-based motor insurance pricing with behavioural data. Annals of Actuarial Science, 13(2), 378–399.CrossRefGoogle Scholar
Durbin, J. (1960) The fitting of time-series models. Revue de l’Institut International de Statistique, 28(3), 233–244.CrossRefGoogle Scholar
Emms, P. and Haberman, S. (2005) Pricing general insurance using optimal control theory. ASTIN Bulletin: The Journal of the IAA, 35(2), 427–453.CrossRefGoogle Scholar
Frees, E.W. (2004) Longitudinal and Panel Data: Analysis and Applications in the Social Sciences, Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Gao, G., Meng, S. and Wüthrich, M.V. (2019) Claims frequency modeling using telematics car driving data. Scandinavian Actuarial Journal, 2019(2), 143–162.CrossRefGoogle Scholar
Hansen, L.P. (1982) Large sample properties of generalized method of moments estimators. Econometrica, 50(4), 1029–1054.CrossRefGoogle Scholar
Henriet, D. and Rochet, J.-C. (1986) La logique des systèmes bonus-malus en assurance automobile: une approche théorique. Annales d’Économie et de Statistique, 1, 133–152.CrossRefGoogle Scholar
Lemaire, J. (2013). Automobile Insurance: Actuarial Models, Vol. 4, New York: Springer.Google Scholar
Levinson, N. (1946) The Wiener (root mean square) error criterion in filter design and prediction. Studies in Applied Mathematics, 25(1–4), 261–278.Google Scholar
Liang, K.-Y. and Zeger, S.L. (1986) Longitudinal data analysis using generalized linear models. Biometrika, 73(1), 13–22.CrossRefGoogle Scholar
Lu, Y. (2018). Dynamic frailty count process in insurance: A unified framework for estimation, pricing, and forecasting. Journal of Risk and Insurance, 85(4), 1083–1102.CrossRefGoogle Scholar
Nelder, J.A. and Wedderburn, R.W. (1972) Generalized linear models. Journal of the Royal Statistical Society: Series A (General), 135(3), 370–384.CrossRefGoogle Scholar
Pinquet, J., Guillén, M. and Bolancé, C. (2001) Allowance for the age of claims in bonus-malus systems. ASTIN Bulletin, 31(2), 337–348.CrossRefGoogle Scholar
Purcaru, O. and Denuit, M. (2003) Dependence in dynamic claim frequency credibility models. ASTIN Bulletin: The Journal of the IAA, 33(1), 23–40.CrossRefGoogle Scholar
Taylor, G.C. (1986) Underwriting strategy in a competitive insurance environment. Insurance: Mathematics and Economics, 5(1), 59–77.Google Scholar
Wüthrich, M.V. (2017) Covariate selection from telematics car driving data. European Actuarial Journal, 7(1), 89–108.CrossRefGoogle Scholar
Supplementary material: PDF

Pinquet supplementary material

Pinquet supplementary material 1

Download Pinquet supplementary material(PDF)
PDF 1.1 MB
Supplementary material: PDF

Pinquet supplementary material

Pinquet supplementary material 2

Download Pinquet supplementary material(PDF)
PDF 748.1 KB
Supplementary material: File

Pinquet supplementary material

Pinquet supplementary material 3

Download Pinquet supplementary material(File)
File 44 KB