math concept
6 topics use this
Math concept
Survival Analysis
Core equation
$$S(t) = P(T > t) = \exp\!\left(-\int_0^t \lambda(u)\,du\right)$$
Survival analysis studies the time until an event — death, failure, relapse. The hazard function, survival function, and censoring are its core concepts, connecting biostatistics, actuarial science, reliability engineering, and economics.
The hazard and survival functions
The hazard rate $\lambda(t) = \lim_{dt\to0} P(T \in [t,t+dt) \mid T \geq t)/dt$ — the instantaneous event rate.
The survival function $S(t) = P(T > t) = \exp(-\Lambda(t))$ where $\Lambda(t) = \int_0^t\lambda(u)\,du$ is the cumulative hazard.
The relationship: $f(t) = \lambda(t)S(t)$, $S(t) = 1 - F(t)$.
Censoring
Right censoring: we observe $\min(T, C)$ where $C$ is a censoring time. The observed data are $(t_i, \delta_i)$ with $\delta_i = \mathbf{1}[T_i \leq C_i]$.
Censoring is crucial: ignoring it (treating censored times as event times) biases estimates downward.
Kaplan-Meier estimator
The non-parametric estimator of $S(t)$:
\[\hat{S}(t) = \prod_{t_i \leq t} \left(1 - \frac{d_i}{n_i}\right)\]where $d_i$ events occur among $n_i$ at-risk individuals at time $t_i$.
Fields that use this concept
Finance & economics
Actuarial science
Actuarial Present Value
The probability-weighted present value of future insurance benefits or annuity payments, combining mortality probabilities with financial discounting.
Life Tables
A systematic tabulation of mortality experience showing the probability of death and survival at each age, forming the foundation of actuarial calculations.
Survival Models in Actuarial Science
Probabilistic models for the future lifetime of individuals and groups, including joint life, multiple decrement, and copula-based dependent mortality models used in pension and life insurance valuation.
Life sciences
Biostatistics
Cox Proportional Hazards
Semi-parametric regression model for survival data that estimates covariate effects without specifying the baseline hazard.
Kaplan-Meier Estimator
The product-limit non-parametric estimator of the survival function, the universal starting point for time-to-event analysis.
Survival Analysis
Statistical methods for time-to-event data, accounting for censoring and the hazard of an event occurring over time.