# Parametric distribution for time to event data - where event is 'uncertain'

Is there a canonical approach to deal with the modeling of time to event ($A$) where $P(A) \ne 1$. For instance, assume the marriage rate is 50%. The study is a set of times (ages) until marriage which may be censored by death. It will also include (roughly - assuming its a longitudinal study in which objects are surveyed under a predetermined time horizon, say 50 years) 50% observations whom will not get married. These observations will be treated as right censored as we are uncertain about the future of these individuals. Consequently, we may end up with a KM estimator which decays with a decreasing rate until ~0.5 where it is becoming flatter.

How could one assume a parametric distribution for this? Most often one uses the theory of survival analysis for these types of problems, but all the distributions (for instance exponential, weibull etc) all assume $P(A)=1$ for the whole lifespan, and would get a poor fit to the data as the empirical distribution does not converge towards 0.

I've been looking through papers and books in the field but cannot understand how this is dealt with. Most examples are select to have a 'fairly' good fit to the classical distributions. Could someone guide me towards literature on the topic or keywords for me to search for them myself.

The Gompertz distribution ($\lambda > 0$, $\rho \neq 0$) has survival function that can written as $$S(t) = \exp \left\{ - \frac{\lambda}{\rho} \left[\exp(\rho t) - 1\right] \right\}$$ Note that if $\rho < 0$ then $$\lim_{t \rightarrow \infty} S(t) = \exp \left( \frac{\lambda}{\rho} \right) > 0$$

so that not everyone experiences the event under study.

This situation is discussed in more detail in the so-called cure models literature.

Gompertz survival functions with $\lambda = 1$:
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