Quantifying the regression to the mean effect in Poisson processes

Regression to the mean (RTM) can occur whenever an extreme observation is selected from a population and a later observation is closer to the population mean. A consequence of this phenomenon is that natural variability can be mistaken as real change. Simple expressions are available to quantify RTM when the underlying distribution is bivariate normal. However, there are many real-world situations, which are better approximated as a Poisson process. Examples include the number of hard disk failures during a year, the number of cargo ships damaged by waves, daily homicide counts in California, and the number of deaths per quarter attributable to acquired immune deficiency syndrome in Australia. In this paper, we derive expressions for quantifying RTM effects for the bivariate Poisson distribution for both the homogeneous and inhomogeneous cases. Statistical properties of our derivations have been evaluated through a simulation study. The asymptotic distributions of RTM estimators have been derived. The RTM effect for the number of people killed in road accidents in different regions of New South Wales (Australia) is estimated using maximum likelihood.