= Solution
An exact observed event contributes \b[its event-time density]:
$$
\boxed{L_i=f(x_i)=-F'(x_i).}
$$
For <exponential-rate estimation from censored exposure>, the event-model <survival likelihood> under <independent censoring> is $\prod_i f(x_i)^{v_i}F(x_i)^{1-v_i}$. For the <exponential distribution>, put $d=\sum_i v_i$ and $X=\sum_i x_i$. Then
$$
L(\theta)=\theta^d e^{-\theta X},\qquad
\ell(\theta)=d\log\theta-\theta X+\text{constant}.
$$
If $d>0$ and $X>0$, $\ell'(\theta)=d/\theta-X$ vanishes at
$$
\boxed{\widehat\theta=\frac{d}{X}.}
$$
The second derivative $-d/\theta^2$ is strictly negative, so this is the unique global <maximum-likelihood estimate>. The likelihood also tends to zero at both ends of the positive parameter range. If $d=0$ and $X>0$, the likelihood decreases on $\theta>0$; its supremum is at $\theta\downarrow0$, with no attained positive estimate. Thus every censored time belongs in the exposure denominator.
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