= Solution
Write $X=\sum_i x_i>0$ and $d=\sum_i v_i$. The exponential <survival likelihood> has $\ell(\lambda)=d\log\lambda-\lambda X$. Its score is $d/\lambda-X$, and for $d>0$ its second derivative is $-d/\lambda^2<0$. Thus the <exponential-rate estimation from censored exposure> gives
$$
\boxed{\widehat\lambda=\frac dX.}
$$
All observation times, including censoring times, belong in $X$. If $d=0$, the <likelihood> decreases with positive $\lambda$ and has its supremum as $\lambda\downarrow0$; under the stated strict positivity constraint there is no interior maximum.
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