Survival analysis as used in the medical context is focused on the concepts of survival function and hazard rate, the latter of these being the basis both for the Cox regression model and of the counting process approach. In spite of apparent simplicity, hazard rate is really an elusive concept, especially when one tries to interpret its shape considered as a function of time. It is then helpful to consider the hazard rate from a different point of view than what is common, and we will here consider survival times modeled as first passage times in stochastic processes. The concept of quasistationary distribution,which is a well-defined entity for various Markov processes, will turn out to be useful.
We study these matters for a number of Markov processes, including the following: finite Markov chains; birth-death processes; Wiener processes with and without randomization of parameters; and general diffusion processes. An example of regression of survival data with a mixed inverse Gaussian distribution is presented.
The idea of viewing survival times as first passage times has been much studied by Whitmore and others in the context of Wiener processes and inverse Gaussian distributions. These ideas have been in the background compared to more popular appoaches to survival data, at least within the field of biostatistics,but deserve more attention.
"Understanding the shape of the hazard rate: a process point of view (With comments and a rejoinder by the authors)." Statist. Sci. 16 (1) 1 - 22, February 2001. https://doi.org/10.1214/ss/998929473