Hazardfunksjonen

Hazardfunksjonen, or hazard function, is the instantaneous risk of a failure or event at time t, conditional on having survived up to t. It is central to survival analysis and reliability theory, used to describe how the risk of an event evolves over time.

Mathematically, for a continuous time-to-event variable T with density f and survival function S, the hazard at time t is h(t) = lim_{dt→0} P(t ≤ T < t+dt | T ≥ t) / dt = f(t) / S(t). The cumulative hazard is H(t) = ∫0^t h(u) du, and the survival function can be written as S(t) = exp(-H(t)). The PDF can be recovered via f(t) = h(t) S(t). In discrete time, h(t) = P(T = t | T ≥ t).

Common examples: for an exponential distribution with constant hazard λ, h(t) = λ; for a Weibull distribution with shape parameter p and scale λ, h(t) = λ p t^{p-1}. In regression settings, the Cox proportional hazards model expresses h(t|X) = h0(t) exp(β'X), where h0 is a baseline hazard and X are covariates.

Interpretation and uses: the hazard provides the instantaneous risk, not a probability, and can increase, decrease, or remain constant over time. It is used in medicine to study time-to-event outcomes, in epidemiology, and in engineering for reliability and risk assessment. Limitations include dependence on the time origin and the fact that different distributions can yield similar hazard behavior.