Survival Analysis: Hazard Functions and Censoring for MAS-I

Survival and Hazard Functions

Survival analysis models time-to-event data. The survival function S(t) = P(T > t) gives the probability of surviving beyond time t. The hazard function h(t) = f(t)/S(t) gives the instantaneous failure rate. The cumulative hazard H(t) = -ln(S(t)). Key relationships: S(t) = exp(-H(t)) and f(t) = h(t)*S(t). Common parametric models include exponential (constant hazard), Weibull (monotone hazard), and log-logistic (non-monotone hazard). The proportional hazards model h(t|x) = h_0(t)*exp(beta'x) separates baseline hazard from covariate effects.

Handling Censored Data

Right censoring occurs when observation ends before the event. The Kaplan-Meier estimator handles censored data: S-hat(t) = product over t_i <= t of (1 minus d_i/n_i), where d_i is events and n_i is the risk set at time t_i. The Nelson-Aalen estimator gives H-hat(t) = sum of d_i/n_i. The log-rank test compares survival between groups. For actuarial applications, survival analysis applies to policyholder lapse studies, time-to-claim analysis, and mortality investigations. Exam MAS-I tests KM estimation, log-rank tests, and Cox regression interpretation.