= Doubly censored data
In the usual left-and-right-censoring sense, <doubly censored data> contain some <left-censored> and some <right-censored> observations, possibly alongside exact values. The record identifies which side is censored. This is the convention used for datasets mixing both sides of censoring.<Nonparametric Estimation of a Survivorship Function with Doubly Censored Data by Bruce W. Turnbull>[Publisher abstract: some response times are censored on the left and some on the right]{ref}.
For known observation limits $L<R$, one can record $Y=\min\{\max(T,L),R\}$ with an indicator specifying $T\le L$, an exact interior event, or $T\ge R$. For a continuous event-time <probability distribution> with <cumulative distribution function> $F$, <probability density function> $f$ and <survival function> $S$, the respective <likelihood contributions> are $F(L)$, $f(T)$ and $S(R)$. A measurement instrument that reports “below its lower detection limit” or “above its upper detection limit” produces precisely this distinction.
If the record only says that $T$ lies outside $(L,R)$ without identifying the side, its contribution would be $F(L)+S(R)$. That is different recorded information and is not the defining convention above. Terminology also needs context when both the origin and endpoint of a duration are uncertain; an <interval-censored> origin together with an <interval-censored> endpoint describes a different observation scheme from simple left-and-right <censoring> of one event time.
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