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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
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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

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