= Solution
Assume <independent censoring>: the censoring mechanism contributes no factor involving the lifetime rate $\theta$. The exponential density and <survival function> are $f_\theta(x)=\theta e^{-\theta x}$ and $S_\theta(x)=e^{-\theta x}$. An observed event contributes the density; a <right-censored> lifetime contributes the <probability> of surviving its censoring time. Thus the <likelihood> for $\theta$, up to censoring factors independent of it, is
$$
L(\theta)=\prod_{i=1}^n f_\theta(x_i)^{v_i}S_\theta(x_i)^{1-v_i}
=\theta^D e^{-\theta T},\qquad
D=\sum_i v_i,\quad T=\sum_i x_i.
$$
For $D>0$ and $T>0$, $\ell'(\theta)=D/\theta-T$ vanishes at $D/T$, and $\ell''(\theta)=-D/\theta^2<0$ proves the maximum:
$$
\boxed{\widehat\theta=\frac{\text{number of observed events}}{\text{total observed person-time}}
=\frac{\sum_i v_i}{\sum_i x_i}.}
$$
Censored individuals add follow-up time to the denominator but no event to the numerator. If $D=0$ and $T>0$, the <likelihood> decreases for $\theta>0$ and has only a supremum as $\theta\downarrow0$; zero is an extended boundary estimate, not a positive-rate exponential <MLE>. The derivation is for independent individuals entering at time zero; delayed entry would require conditional survival contributions and exposure measured from entry.
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