Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 35 / 4 / a

Under independent censoring, observation of an event-free individual to $x_i$ contributes the survival probabilityHere $F$ denotes the survivor function as in the paper, rather than a cumulative distribution function. The censoring distribution contributes an additional factor to the full observed-data likelihood, but if it has distinct parameters and the required independence holds, that factor can be omitted when estimating the event-time parameters. A censored observation still conveys that the event time exceeded $x_i$.

An exact observed event contributes its event-time density:For exponential-rate estimation from censored exposure, the event-model survival likelihood under independent censoring is ${\prod }_{i}f(x_i{)}^{v_i}F(x_i{)}^{1-v_i}$. For the exponential distribution, put $d={\sum }_i{v}_i$ and $X={\sum }_i{x}_i$. ThenIf $d>0$ and $X>0$, ${\ell }^{\prime }(\theta )=d/\theta -X$ vanishes atThe second derivative $-d/{\theta }^2$ is strictly negative, so this is the unique global maximum-likelihood estimate. The likelihood also tends to zero at both ends of the positive parameter range. If $d=0$ and $X>0$, the likelihood decreases on $\theta >0$; its supremum is at $\theta \downarrow 0$, with no attained positive estimate. Thus every censored time belongs in the exposure denominator.