 Solution

For an exponential distribution of rate $\theta >0$, the probability density function and survivor function are $f(t)=\theta e^{-\theta t}$ and $F(t)=e^{-\theta t}$. Setwhere $D$ is the observed event count and $E$ is total person-time at risk. The log-likelihood, up to a constant, becomesWhen $D>0$ and $E>0$, its score function is $D/\theta -E$, which vanishes atIts second derivative is negative, so this is the unique maximum-likelihood estimator. This is exponential-rate estimation from censored exposure: the estimate is observed events divided by all observed exposure, including exposure from right-censored individuals.

If $D=0$ and $E>0$, $\ell =-\theta E$ decreases strictly. There is no maximizer in the open parameter space $\theta >0$; its supremum occurs as $\theta \downarrow 0$, or the extended boundary estimate is $\hat{\theta }=0$ if a zero rate is permitted. The formula should not be interpreted as an interior regular fit in this case.