= Solution
For an <exponential distribution> of rate $\theta>0$, the <probability density function> and <survivor function> are $f(t)=\theta e^{-\theta t}$ and $F(t)=e^{-\theta t}$. Set
$$
D=\sum_i v_i,\qquad E=\sum_i x_i,
$$
where $D$ is the observed event count and $E$ is total <person-time at risk>. The <log-likelihood>, up to a constant, becomes
$$
\ell(\theta)=D\log\theta-\theta E.
$$
When $D>0$ and $E>0$, its <score function> is $D/\theta-E$, which vanishes at
$$
\boxed{\widehat\theta=\frac{D}{E}.}
$$
Its 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 $\theta\downarrow0$, or the extended boundary estimate is $\widehat\theta=0$ if a zero rate is permitted. The formula should not be interpreted as an interior regular fit in this case.
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