The Predictive Distribution for a Poisson Claims Model by Glen Barnett and Angela Tong

We give a derivation of the frequentist predictive distribution for a simple Poisson model, show how this simple model may be adapted to fit a claim count triangle, and demonstrate how the predictive distribution of aggregates of future claim counts may be computed for the adapted model. Quantiles and prediction intervals are readily obtained. Quasi-Poisson models for incremental paid losses are commonly used in reserving and risk capital calculations. The results in this paper may readily be extended to that situation for a similar model to the one applied to claim counts. Section 1: Introduction This section describes two models – a very simple Poisson count with exposure model with a single parameter, which we extend to a more complex illustrative example, and secondly a model for a claim count triangle that depends on multiple Poisson parameters. The result for the predictive distribution for the Poisson may be applied to these situations, and quantiles of individual and aggregate forecasts may be obtained. 1.1 Poisson counts with known exposures Consider the case where we have a set of Poisson counts with known exposures and a common Poisson rate per unit of exposure. That is, consider a vector of independent Poisson random variables X1, ..., Xn, representing a set of counts with known exposures k1 , ..., kn. Xi ~ Poisson(μi ) where μi = ki · λ. While λ is unknown, we can estimate it from the observations on the X’s, x1, x2, ..., xn. We wish to obtain the predictive distribution of and additional observation, Xn+1 (with known exposure kn+1), conditional on the observed Poisson counts (x1, ..., xn) and their exposures. 1.2 An illustrative example Single-vehicle accidents occurring on non-holiday weekdays on two different stretches of a highway are recorded for several months. One stretch is long (length dL), and the second is short (dS). Data is observed for n time periods, of length (in number of non-holiday-weekdays) t1, t2, ..., tn. Interest focuses on prediction of the number of single-vehicle accidents in the next time period (length tn+1) for both stretches, and on the combined total across both stretches of highway.