Hazard function

Summary

The hazard function of a lifetime $T$ gives the instantaneous rate of the event at time $t$. The rate is taken among the subjects that have not had the event yet. It equals the density of $T$ divided by its survival function. Its integral, the cumulative hazard, determines the survival function through $S(t) = e^{-H(t)}$. Hazards are the natural scale of many models in survival analysis, such as the proportional hazards model. In reliability theory the hazard is called the failure rate.