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Bruce W. Turnbull works on survival analysis and the statistical monitoring of clinical trials. His Cornell profile connects research on statistical methods with the practical problem of deciding what accumulating trial data justify.^[ref]

His 1974 paper studied nonparametric estimation when event times are subject to both left censoring and right censoring. A time known only to precede a threshold and one known only to exceed a threshold convey different information about the same survival function. Retaining those inequalities makes incomplete observations useful without substituting an invented event time.^[ref]

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.^[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.