Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 46 / 3 / b

Under independent censoring conditional on the modelled covariates, an observed event contributes $f_i(x_i)=h_i(x_i)e^{-H_i(x_i)}$, and a right-censored observation contributes $S_i(x_i)=e^{-H_i(x_i)}$. Factors from the censoring distribution may be omitted when they contain no survival-model parameters. The survival likelihood and log-likelihood are thereforeCensored individuals still contribute their observed exposure through $H_i(x_i)$; they are not deleted from the analysis.

Put $d={\sum }_i{v}_i$. With common Weibull distribution parameters and the hazard from part (a), substitution into the general log-likelihood givesThis is up to censoring-mechanism terms independent of $\lambda ,p$. For $d>0$ and fixed $p$, its maximizing rate is $(d/{\sum }_i{x}_i^p{)}^{1/p}$; the shape still requires separate estimation if it is unknown.

Write $X={\sum }_i{x}_i>0$ and $d={\sum }_i{v}_i$. The exponential survival likelihood has $\ell (\lambda )=d\log \lambda -\lambda X$. Its score is $d/\lambda -X$, and for $d>0$ its second derivative is $-d/{\lambda }^2<0$. Thus the exponential-rate estimation from censored exposure givesAll observation times, including censoring times, belong in $X$. If $d=0$, the likelihood decreases with positive $\lambda$ and has its supremum as $\lambda \downarrow 0$; under the stated strict positivity constraint there is no interior maximum.

Set $g_i=G_i^{\prime }(x_i)$, $X={\sum }_i{x}_i$, $d={\sum }_i{v}_i$ and $A={\sum }_i{v}_i{g}_i$. Then $h_i(x_i)=\lambda +g_i$ andThe integrated offsets are known and enter only as constants in estimation of $\lambda$. Let ${\lambda }_0=d/X$ and assume $d>0$ and every $g_i$ at an event is small compared with ${\lambda }_0$. Expanding the score givesNow write $\lambda ={\lambda }_0+\delta$ and retain first-order terms in $g_i,\delta$. Since $d/{\lambda }_0=X$, the equation becomes $-d\delta /{\lambda }_0^2-A/{\lambda }_0^2=0$. Therefore the exponential rate with a small known additive hazard estimate isThe correction is negative: some event risk is already supplied by the known additive hazards. As a check, if every $g_i=g$, the score equation gives the exact positive interior estimate $d/X-g$. The approximation is intended for small offsets and a positive interior solution; a negative approximation or no observed events requires handling the boundary rather than reporting a negative rate.