Parametric survival and reliability models

Let $T$ be a nonnegative event time as in Event-time observation, censoring, and truncation schemes, with law $P_T$. Its cumulative distribution function is $F_T(t)=P(T\le t)$ (Cumulative distribution function of a real random variable). The survival function is $S_T(t):=P(T>t)=1-F_T(t),t\ge 0.$ Write $S_T(t^{-}):=P(T\ge t)$ for the pre-event survival. In particular, $S_T(0)=1$ exactly when $P(T=0)=0$.

For a nonrepairable unit, meaning that a failed unit is permanently removed from the population, its reliability function is $R_T(t):=P(T>t)=S_T(t)$. This is a single first-failure lifetime model. A repairable system can fail repeatedly; its cumulative failure count and rate of occurrence of failures are not summarized by this one-lifetime survival function.

When the law of $T$ has a specified density $f$ relative to Lebesgue measure, meaning $P_T(A)={\int }_{A}f(u)du$ for each Borel set $A$, and $S_T(t)>0$, its density hazard is the chosen-version function ${\lambda }_T(t):=\frac{f(t)}{S_T(t)}\mathrm{.}$ Values can change on a Lebesgue-null set when the density representative changes. Where this hazard is integrable, the cumulative hazard is ${\mathrm{\Lambda }}_T(t):={\int }_0^t{\lambda }_T(u)du\mathrm{.}$ For a law that may have an atom at $t$, its jump hazard is $q_T(t):=\frac{P(T=t)}{P(T\ge t)}$ when $P(T\ge t)>0$. The jump hazard is a conditional event probability, rather than a density per unit time.

For parameters $\lambda >0$, the exponential survival specification is $S_T(t)=e^{-\lambda t}$ for $t\ge 0$; the family of laws is denoted $T\sim \mathrm{E}\mathrm{x}\mathrm{p}(\lambda )$. For shape $k>0$ and scale $\eta >0$, the Weibull survival specification is $S_T(t)=e^{-(t\mathrm{/}\eta {)}^k}$ for $t\ge 0$. Under Countable Choice, these are parameter-indexed families of sampling laws in the sense of Parametric statistical models; their existence and densities are established in Exponential and Weibull survival specifications define probability laws ↗.

The density-hazard identity for absolutely continuous event-time laws and the jump relation for atoms are separate results. An empirical pattern such as a decreasing-then-increasing “bathtub” hazard is not a consequence of these definitions or of either parametric family.