Smoothing dispersed counts with applications to mortality data

Mortality data are often classified by age at death and year of death. This classification results in a heterogeneous risk set and this can cause problems for the estimation and forecasting of mortality. In the modelling of such data, we replace the classical assumption that the numbers of claims follow the Poisson distribution with the weaker assumption that the numbers of claims have a variance proportional to the mean. The constant of proportionality is known as the dispersion parameter and it enables us to allow for heterogeneity; in the case of insurance data the dispersion parameter also allows for the presence of duplicates in a portfolio. We use both the quasi-likelihood and the extended quasi-likelihood to estimate models for the smoothing and forecasting of mortality tables jointly with smooth estimates of the dispersion parameters. We present three main applications of our method: first, we show how taking account of dispersion reduces the volatility of a forecast of a mortality table; second, we smooth mortality data by amounts, ie, when deaths are amounts claimed and exposed-to-risk are sums assured; third, we present a joint model for mortality by lives and by amounts with the property that forecasts by lives and by amounts are consistent. Our methods are illustrated with data from the Continuous Mortality Investigation.