This paper considers a particular credibility model for the claim numbers N1, 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 claim amounts have 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 of Ni given the underlying risk parameter does not vary with i.

The computation of the credibility estimate of Nn+1 involves the considerable labor of inverting an n × n covariance matrix (n is 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. Typically nth order stationary a priori sequences (e.g., ARMA (p, q)-processes) lead to an nth order recursive scheme. In this paper we impose the restriction that the conditional distribution of Ni is 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 an arbitrary weakly stationary a priori sequence. 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) as a priori sequence and models which can be considered as (discrete) generalizations of the Pólya process.

Recursive credibility estimation is discussed from the viewpoint of linear filtering theory. A conjunction of geometric interpretation and the innovation approach leads to general algorithms not developed before. Moreover, covariance characterizations considered by other researchers drop our elegantly as a result of geometric considerations. Examples are presented of Kalman type filters valid for non-Gaussian measurements.

The unbayesed credibility procedure proposed by Gerber is revisited. Its performance is discussed, connections are drawn to earlier literature, and some possible ideas of generalizations are investigated (and found fruitless).

We improve on some results of Sundt (1982) on the asymptotic behaviour of compound negative binomial distributions.

This paper considers reinsurance retention limits in cases where the cedent has a choice between a pure quota-share treaty, a pure excess of loss treaty or a combination of the two. Our primary aim is to find the combination of retention limits which minimizes the skewness coefficient of the insurer's retained risk subject to constraints on the variance and the expected value of his retained risk. The results are given without specifying precisely how the excess of loss reinsurance premium is calculated. It is also shown that, depending to some extent on the constraint on the variance, the solution to the problem is a pure excess of loss treaty if the excess of loss premium is calculated using the expected value or standard deviation principle but that this need not be true if the variance principle is used.