Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 44 / 1

The paper uses $F$ for the survivor function, rather than for the cumulative distribution function. For a nonnegative survival time $T$ with an absolutely continuous probability distribution, defineWhere $F(t)>0$, the hazard function is the instantaneous event rate conditional on survival:The integrated hazard is $H(t)={\int }_0^{t}h(u)du$. Since $h=-F^{\prime }/F$, integration gives $H(t)=\log F(0)-\log F(t)$. With the usual time origin and no event at zero, $F(0)=1$, soConversely, differentiating $F=e^{-H}$ gives $F^{\prime }=-hF$, so these relationships determine each of the survivor function, hazard function and integrated hazard from either of the others. A positive limiting survivor function is possible if the integrated hazard remains finite; a proper finite event time instead has $F(t)\to 0$.

Assume independent individuals and independent censoring: conditional on any modelled covariates, the censoring mechanism gives no additional information about the latent event time and has no parameter shared with its survival distribution. An observed event at $x_i$ contributes the probability density function $f(x_i)$. A right-censored observation contributes the survivor function $F(x_i)$, since it establishes only $T_i>x_i$.

More formally, let $C_i$ be an independent censoring time with density $c_i$ and survivor function $G_i$. The observed pair $X_i=\min (T_i,C_i)$, $v_i=1_{\{T_i\le C_i\}}$ has a parameter-dependent factor $f(x_i)G_i(x_i)$ if $v_i=1$ and $F(x_i)c_i(x_i)$ if $v_i=0$. The factors involving only the censoring distribution can be omitted from the event-time likelihood. ThusUsing $f=hF$ also gives the useful survival likelihood formEvery individual therefore contributes survival exposure, whether or not an event is observed. These formulas do not justify ignoring informative censoring; that situation requires a model for the observation mechanism as well.

For an exponential distribution of rate $\theta >0$, the probability density function and survivor function are $f(t)=\theta e^{-\theta t}$ and $F(t)=e^{-\theta t}$. Setwhere $D$ is the observed event count and $E$ is total person-time at risk. The log-likelihood, up to a constant, becomesWhen $D>0$ and $E>0$, its score function is $D/\theta -E$, which vanishes atIts second derivative is negative, so this is the unique maximum-likelihood estimator. This is exponential-rate estimation from censored exposure: the estimate is observed events divided by all observed exposure, including exposure from right-censored individuals.

If $D=0$ and $E>0$, $\ell =-\theta E$ decreases strictly. There is no maximizer in the open parameter space $\theta >0$; its supremum occurs as $\theta \downarrow 0$, or the extended boundary estimate is $\hat{\theta }=0$ if a zero rate is permitted. The formula should not be interpreted as an interior regular fit in this case.

Differentiating the log-likelihood again givesThus the observed Fisher information for the rate is proportional to the total number of observed events. At a positive interior maximum-likelihood estimator, the large-sample variance estimate and standard error arePrecision is therefore driven by the number of observed events rather than by the number recruited alone. Heavy right censoring can leave a large study with little information about its hazard function. Under the regularity conditions for the Fisher information, the expected information is $\mathbb{E}[D]/{\theta }^2$.

A right-censored observation is not wholly uninformative: its exposure contributes to $E$, changes the maximum-likelihood estimator, and affects the expected number of events under a sampling design. Its contribution $-\theta x_i$ happens to have zero second derivative in this rate parameterization. For example, if no events occur, the likelihood still declines as $e^{-\theta E}$ and disfavors large rates, despite its zero curvature. The usual interior variance approximation is then invalid.