= 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> are
$$
L(\theta)=\prod_i[\theta e^{-\theta x_i}]^{\Delta_i}[e^{-\theta x_i}]^{1-\Delta_i}=\theta^d e^{-\theta E},\qquad \ell(\theta)=d\log\theta-\theta E.
$$
For $d>0$, the <score function> $\ell'(\theta)=d/\theta-E$ vanishes at $d/E$, and $\ell''(\theta)=-d/\theta^2<0$. This is the unique maximum over $\theta>0$:
$$
\boxed{\widehat\theta=d/E=\widetilde\theta.}
$$
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 $\theta\downarrow0$, or attained at zero if that boundary is admitted.
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