= Solution
Differentiating the <log-likelihood> again gives
$$
\boxed{\ell''(\theta)=-\frac{D}{\theta^2},\qquad
J(\theta):=-\ell''(\theta)=\frac{D}{\theta^2}.}
$$
Thus 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> are
$$
\widehat{\operatorname{Var}}(\widehat\theta)=J(\widehat\theta)^{-1}
=\frac{\widehat\theta^2}{D},\qquad
\operatorname{se}(\widehat\theta)\simeq\frac{\widehat\theta}{\sqrt D}.
$$
Precision 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.
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