In many circumstances, the risk of death increases with age. There are many models to chose from:
The Rayleigh is obviously the simplest model in which hazard increases with time, but the Makeham is widely regarded as the best description of hazard for human subjects. After infancy, there is a constant hazard (a) which is due to age-independent accidents, murder, suicides, etc., with an exponentially increasing hazard in later life. The Gompertz assumption was that ‘the average exhaustion of a man's power to avoid death is such that at the end of equal infinitely small intervals of time he has lost equal portions of his remaining power to oppose destruction which he had at the commencement of these intervals’. Note that the Gompertz differs from the Makeham only by the omission of the extra background hazard (a), and this becomes negligible in old age.
These plots show how hazard changes with age for the following distributions: from top left to bottom right: exponential, Weibull, Gompertz, Makeham, extreme value and Rayleigh.
There are three cases that concern us here:
The first case is dealt with very simply in R by ...