 Random relabelling is not independent right censoring

Suppose the recorded durations retain an exponential distribution of rate $\theta$, but an independent fraction $q$ are relabelled as failures and the rest as right-censored. Applying the ordinary survival likelihood gives $\hat{\theta }=d/{\sum }_i{y}_i\to q\theta$, rather than $\theta$. A censoring flag chosen independently of the recorded duration does not establish independent censoring of a latent failure time. Genuine simulation instead draws an event time and an independent censoring time and records their minimum.