mathematics//statistics//survival analysis//hazard function

The hazard function is a function of time that gives the instantaneous risk that an event happens right now, given that it has not happened yet, and it is the core object of survival analysis; in reliability engineering it is the **failure rate**, and it answers the question maintenance turns on: does an old unit fail more readily than a new one? With \(f(t)\) the density of the time to the event and \(R(t)\) the fraction still surviving,

The hazard function is a function of time that gives the instantaneous risk that an event happens right now, given that it has not happened yet, and it is the core object of survival analysis; in reliability engineering it is the failure rate, and it answers the question maintenance turns on: does an old unit fail more readily than a new one? With f(t)f(t)f(t) the density of the time to the event and R(t)R(t)R(t) the fraction still surviving,

h(t)=f(t)R(t),h(t)=\frac{f(t)}{R(t)},h(t)=R(t)f(t)​,

so that h(t) Δth(t),\Delta th(t)Δt is the probability of failing in the next short interval Δt\Delta tΔt for a unit that has reached age ttt. Dividing by the survivors is the whole point: the density says how many of the original population die around ttt, the hazard says how dangerous that age is for those still alive.

Its shape is read directly as a diagnosis of the failure mechanism. A falling hazard means early defects being weeded out, a constant one random shocks, a rising one wear, and the three in sequence draw the bathtub curve. A constant hazard is the memoryless case of the exponential distribution: a unit with 10,000 hours is as likely to fail tomorrow as one installed today, and replacing it early only spends spares (preventive maintenance). The Weibull distribution covers all three shapes with one parameter.

It turns a point process into a per-window risk that can be forecast: the intensity of a point process is a hazard for the next event.

It separates predicting that a system is fragile from predicting the exact moment it breaks. Most risk models do the first; timing is the hard part, and for one unit it needs that unit's own measurements (remaining useful life).

A high hazard describes the distribution and says nothing certain about the next draw: a loaded die is not an oracle.