Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 40 / 1 / b / iv / Solution

For independent individuals under fixed administrative censoring, an observed failure contributes the probability density function $\theta e^{-\theta x_i}$, while a censored observation contributes the survivor function $e^{-\theta x_i}$. Hence the survival likelihood and log-likelihood areFor $d>0$, the score function ${\ell }^{\prime }(\theta )=d/\theta -E$ vanishes at $d/E$, and ${\ell }^{\prime \prime }(\theta )=-d/{\theta }^2<0$. This is the unique maximum over $\theta >0$:Thus maximum likelihood estimation agrees with the observed-exposure estimating equation. If there are no failures, the likelihood decreases with $\theta$: its supremum is approached as $\theta \downarrow 0$, or attained at zero if that boundary is admitted.