Project Euclid

When making probabilistic models for survival times, one should consider the fact that individuals are heterogeneous. The observed changes in population intensities (or hazard rates) over time are a mixed result of two influences: on the one hand, the actual changes in the individual hazards, and, on the other hand, the selection due to high-risk individuals leaving the risk group early. I will consider the common multiplicative model for heterogeneity, but with the new feature that the random proportionality factor has a compound Poisson distribution. This distribution is studied in some detail. It is pointed out how its application to the survival situation extends a model of Hougaard, inheriting several nice properties. One important feature of the model is that it yields a subgroup of zero susceptibility, which "survives forever." This is a relevant model in medicine and demography. Two examples are given where the model is fitted to data concerning marriage rates and fertility.