Hazardraten

Hazardraten, commonly referred to in English as hazard rates, are measures of the instantaneous risk that a particular event occurs at a given time, assuming the event has not yet happened. They are used in survival analysis, reliability engineering, and related fields to describe how quickly events such as death, failure, or default occur over time.

Mathematically, the hazard rate h(t) is defined as the instantaneous risk of the event at time t given survival up to t. A standard relation links the hazard to the probability density f(t) of the event time T and the survival function S(t) = P(T > t): h(t) = f(t) / S(t). The survival function and the hazard are connected through the cumulative hazard H(t) = ∫0^t h(u) du and S(t) = exp(-H(t)) for continuous time.

Interpretation of hazard rates varies by context. A constant hazard implies an exponential time-to-event distribution. Increasing hazard suggests that risk grows with time or age (aging or wear-out), while decreasing hazard can indicate early failures or improvements over time. In finance and credit risk, hazard rates approximate the instantaneous probability of default given survival to time t.

Estimation and modeling approaches include nonparametric methods like the Kaplan-Meier estimator for survival probabilities and the Nelson-Aalen estimator for the cumulative hazard. Parametric models (for example, Weibull or Gompertz) specify a functional form for h(t); semi-parametric models such as the Cox proportional hazards model estimate hazard ratios without fully specifying h(t).

Applications of Hazardraten span clinical prognosis, insurance and pension risk, and reliability testing. Limitations include sensitivity to censoring mechanisms and the need for correct model assumptions, as the hazard rate reflects instantaneous risk rather than a straightforward probability.