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

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.

Condition on the bulb being alive at age $\tau$. The next hour's survival likelihood contribution isIt does not depend on the unknown age $\tau$. This is memorylessness of the exponential distribution: the residual lifetime given survival has the same exponential distribution as a new lifetime.

For an exact failure half an hour after entry, use the conditional density, rather than the probability of an event at one exact time:Again this is independent of $\tau$ by memorylessness of the exponential distribution. Whether successive observed lecture segments came from the same bulb does not affect the conditional likelihood: surviving segments of one bulb multiply into its survival exposure, and replacement starts another memoryless lifetime. Assume no unrecorded use relevant to the exposure calculation or, equivalently, condition on the bulb being alive at each observed lecture's start; unobserved ages themselves need not be known.

The observed working exposure is $12+1/2=12.5$ bulb-hours with one failure. ThusStrict concavity of $\log \lambda -12.5\lambda$ verifies the maximum. Immediate replacement would increase observed working exposure to 13 hours, since the replacement survives the remaining half hour, givingFor the alternative Weibull distribution and unknown entry age in a Weibull survival segment, keep its different inverse-scale convention: $F(t)=\exp [-(\lambda t{)}^p]$. Conditional survival for an additional duration $u$ isand the conditional failure density after $u$ isBoth depend on the unknown age unless $p=1$. The available segment data, with ages and bulb identity unrecorded, do not specify the usual Weibull likelihood needed to estimate $\lambda ,p$. Merely collecting more of the same unknown-age records does not fix the missing age/design information. A larger dataset with documented ages or identities, or a justified model for the entry-age distribution, could permit estimation. If $p=1$ is imposed, the exponential calculation remains valid. The issue is loss of memorylessness and unspecified entry history, not a universal impossibility of learning from any larger bulb experiment.