Association Behaviour in Non-Linear Death Processes

Transition probabilities within a birth and death process are difficult to find analytically. As a consequence pure birth and pure death processes are considered in seperate systems and it is found that both systems can be fully specified from intensity specifications. In a linear continuous-time Markovian pure death process it is found that optional death times are independent. Faddy (1985) conjectured that concave (convex) death rates μ0, μ1, ..., μN exhibit increasing (decreasing) variability compared to the linear case. Ball and Donnelly (1987) claim that the conjecture is true and their result gives circumstances in which the covariance, and thus the correlation, is increased (decreased) relative to the linear case. Brown and Donnelly (1993) find an error in Ball and Donnelly (1987) as an aspect of their proof cannot be used to relate death intensities to conditional probabilities. Lefèvre and Milhaud (1990) take advantage of the conditional independence structure and produce a general result which relates individual death rates μ1, μ2 2 , ..., μn n to interparticle correlation and covariance configuration. This is proved by Brown and Donnelly (1993). The interparticle correlation results are applied to life data and it is found that male and female death times are positively correlated.