An IBNYR event is one that occurs randomly during some fixed exposure interval and incurs a random delay before it is reported. A previous paper developed a continuous-time model of the IBNYR process in which both the Poisson rate at which events occur and the parameters of the delay distribution are unknown random quantities; a full-distributional Bayesian method was then developed to predict the number of unreported events. Using a numerical example, the success of this approach was shown to depend upon whether or not the occurrence dates were available in addition to the reporting dates. This paper considers the more usual practical situation in which only discretized epoch information is available; this leads to a loss of predictive accuracy, which is investigated by considering various levels of quantization for the same numerical example.

An IBNYR event is one that occurs randomly during some fixed exposure interval and incurs a random delay before it is reported. Both the rate at which such events occur and the parameters of the delay distribution are unknown random quantities. Given the number of events that have been reported during some observation interval, plus various secondary data on the dates of the events, the problem is to estimate the true values of the unknown parameters and to predict the number of events that are still unreported. A full-distributional Bayesian model is used, and it is shown that the amount of secondary data is critical. A recursive procedure calculates the predictive density; however, an explicit formula for the predictive mode can be obtained. The main computational work is the evaluation of an integral involving the prior density of the delay parameters, but this can be simplified in the exponential case using Gammoid approximations.