MMP.7:5.3 - Use a timed-out trial as an interval report

A test asks how long an event takes. Each independent trial is observed until its event or a fixed timeout c. Record both V=min(T,c) and a flag D indicating whether the event occurred before timeout. As an illustrative subject assumption, let T have exponential density lambda exp(-lambda t) for t at least zero, with lambda positive.

An event at time t before c contributes the density lambda exp(-lambda t). A timeout contributes P(T>=c)=exp(-lambda c), obtained by integrating the density over the unobserved tail. For m completed trials and total observed time S, including the timeout durations, the likelihood is proportional to lambda^m exp(-lambda S).

With completions at times 1 and 2 and one timeout at 4, S=7 and m=2. Maximizing this illustrative likelihood gives lambda_hat=2/7. Treating the timeout as a third event instead gives 3/7; dropping it gives 2/3. The flag determines which operation is correct.

If only completed trials enter a database and neither the number nor identities of timed-out trials are available, the observed-time density instead conditions on completion: divide the event density by 1-exp(-lambda c) on the interval before c. A changed recording rule changes the model even when the stored times look the same. The exponential assumption is dispensable: a different duration law supplies its own event density and tail probability.