Exponential event-time observations with censoring and delayed entry

Integrating the density gives $F_{\lambda }(t)=1-e^{-\lambda t}$ and $S_{\lambda }(t)=e^{-\lambda t}$ for $t\ge 0$; in particular $F_{\lambda }(0)=0$ and $S_{\lambda }(0)=1$.

With fixed censoring time $C=2$, an event observed at $U=1$ has event factor $f_{\lambda }(1)=\lambda e^{-\lambda }$ and censor factor $1$. A censored observation at $U=2$ has event factor $S_{\lambda }(2)=e^{-2\lambda }$ and censor factor $1$ relative to the point-mass censoring law.

An interval observation $1<T\le 2$ has probability $F_{\lambda }(2)-F_{\lambda }(1)=(1-e^{-2\lambda })-(1-e^{-\lambda })=e^{-\lambda }-e^{-2\lambda }$.

At fixed inspection time $C=1$, current status $\delta =1$ has event factor $F_{\lambda }(1)=1-e^{-\lambda }$, and $\delta =0$ has event factor $S_{\lambda }(1)=e^{-\lambda }$.

After delayed entry $L=1$, with fixed censoring time $C=3$, an event observed at $U=2$ has conditional event factor $f_{\lambda }(2)\mathrm{/}{S}_{\lambda }(1)=\lambda e^{-2\lambda }\mathrm{/}{e}^{-\lambda }=\lambda e^{-\lambda }$. A censored observation at $U=3$ has conditional event factor $S_{\lambda }(3)\mathrm{/}{S}_{\lambda }(1)=e^{-2\lambda }$.

If instead the sample is right-truncated at $R=2$, the selected-sample density is $\lambda e^{-\lambda t}\mathrm{/}(1-e^{-2\lambda })$ for $0\le t<2$, since $P(T<2)=F_{\lambda }(2)=1-e^{-2\lambda }$. This is a conditional event-time density, not a right-censoring contribution.

The censoring and truncation examples use different event factors: ordinary right censoring uses $f_{\lambda }(u)$ or $S_{\lambda }(u)$, delayed entry divides by $S_{\lambda }(1)$, and right truncation divides by $P(T<2)$.