An Evolutionary Credibility Model for Claim Numbers
Peter A. Albrecht · Astin Bulletin · 1985
This paper considers a particular credibility model for the claim numbersN1,N2, …,Nn, … of a single risk within a collective in successive periods 1, 2, …,n, … In the terminology of Jewell (1975) the model is an evolutionary credibility model, which means that the underlying risk parameter Λ is allowed to vary in successive periods (the structure function is allowed to be time dependent). Evolutionary credibility models for claimamountshave been studied by Bühlmann (1969, pp. 164–165), Gerber and Jones (1975), Jewell (1975, 1976), Taylor (1975), Sundt (1979, 1981, 1983) and Kremer (1982). Again in Jewell's terminology the considered model is on the other hand stationary, in the sense that the conditional distribution ofNigiven the underlying risk parameter does not vary withi. The computation of the credibility estimate ofNn+1involves the considerable labor of inverting ann×ncovariance matrix (nis the number of observations). The above mentioned papers have therefore typically looked for model structures for which this inversion is unnecessary and instead a recursive formula for the credibility forecast can be obtained. Typicallynth order stationarya priorisequences (e.g., ARMA (p, q)-processes) lead to annth order recursive scheme. In this paper we impose the restriction that the conditional distribution ofNiis Poisson (which by the way leads to a model identical to the so called “doubly stochastic Poisson sequences” considered in the theory of stochastic point processes). What we gain is a recursive formula for the coefficients of the credibility estimate (not for the estimate itself!) in case of anarbitraryweakly stationarya priorisequence. In addition to this central result the estimation of the structural parameters is considered in this case and some more special models are analyzed. Among them are EARMA-processes (which are positive-valued stationary sequences possessing exponentially distributed marginals and the same autocorrelation structure as ARMA-processes) asa priorisequence and models which can be considered as (discrete) generalizations of the Pólya process.