This paper addresses the Bayesian estimation of the shape parameter of Pareto distributions, and its application to premium calculation of large claims excess of loss (XL) reinsurance contracts. It studies the use of the generalized inverse Gaussian (GIG) as a Pareto prior conjugate, a family that contains as a particular case the gamma distribution. An exact credibility formula is deduced allowing the calculation of individual reinsurance premiums. These are premiums suited to the excesses history of a sole portfolio. A family of predictive distributions for the excesses is derived. We apply our exact credibility model to a sample of excesses arisen in ten Spanish portfolios of liability motor insurance from year 1992 to year 2001.

Estimation of rates of onset of rare, late-onset dominantly inherited genetic disorders is complicated by: (a) probable ascertainment bias resulting from the ‘recruitment’ of strongly affected families into studies; and (b) inability to identify the true ‘at risk’ population of mutation carriers. To deal with the latter, Gui & Macdonald (2002a) proposed a non-parametric (Nelson-Aalen) estimate (x) of a simple function Λ(x) of the rate of onset at age x. The function Λ(x) had a finite bound, which was an increasing function of the probability p that a child of an affected parent inherits the mutation and σ the life-time penetrance. However if (x) exceeds this bound, it explodes to infinity, and this can happen at quite low ages. We show that such ‘failure’ may in fact be a useful measure of ascertainment bias. Gui & Macdonald assumed that p = 1/2 and σ = 1, but ascertainment bias means that p > 1/2 and σ ≠ 1 in the sample. The maximum attained by (x) allows us to estimate a range for the product pσ, and therefore the degree of ascertainment bias that may be present, leading to bias-corrected estimates of rates of onset. However, we find that even classical independent censoring, prior to ascertainment, can introduce new bias. We apply these results to early-onset Alzheimer’s disease associated with mutations in the Presenilin-1 gene.

A number of standard market models are studied. For each one, algorithms of computational complexity equal to the number of rates times the number of factors to carry out the computations for each step is introduced. Two new classes of market models are developed and it is shown for them that similar results hold.

The authors review various facts about copulas linking discrete distributions. They show how the possibility of ties that results from atoms in the probability distribution invalidates various familiar relations that lie at the root of copula theory in the continuous case. They highlight some of the dangers and limitations of an undiscriminating transposition of modeling and inference practices from the continuous setting into the discrete one.

It has been known since Zehnwirth (1977) that a scalar credibility coefficient is closely related to the F-statistic of an analysis of variance between and within risk clauses. The F-statistic may also be viewed as testing a certain regression structure, associated with credibility framework, against the null hypothesis of a simpler structure.

This viewpoint is extended to multi-dimensional credibility frameworks in which the credibility coefficient is a matrix (Sections 3 and 4), and to hierarchical regression credibility frameworks (Section 6). In each case the credibility coefficient is expressed in terms of the F-statistic that tests the significance of a defined regression structure against a simpler one.

Section 5 points out how the computation may be implemented in certain cases by means of regression software.