= Solution
Let $C$ be the clump size and put $q=e^{-\lambda t}$. The probability that all its members die, conditional on $C=n$, is $(1-q)^n$. Summing the <geometric series> gives
$$
\mathbb P(\text{clump dead})=\sum_{n\ge1}(1-\theta)\theta^{n-1}(1-q)^n
=\frac{(1-\theta)(1-q)}{1-\theta(1-q)}.
$$
Subtracting from one yields $\boxed{p(t,\lambda)=q/(1-\theta+\theta q)}$, for $0\le\theta<1$.
Assuming independent clumps and independent repeats, the combined <likelihood> is proportional to $p^R(1-p)^{N-R}$. For $0<R<N$ its unique maximizer is $\widehat p=R/N$. The relation between $p$ and $q$ is one-to-one, with $q=(1-\theta)p/(1-\theta p)$. For $t>0$ this gives
$$
\boxed{\widehat\lambda=\frac1t\log\frac{N-\theta R}{R(1-\theta)}}.
$$
If $R=N$, the estimate is the boundary value zero; if strictly positive rates are required, it is only a limiting supremum. If $R=0$, the <likelihood> approaches its supremum as $\lambda\to\infty$ and there is no finite estimate. These boundary cases explain when the printed logarithmic expression is finite.
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