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

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.