Pooled rate estimation for several Poisson processes under Type-I censoring with unrecorded censoring times

I have a pooled estimator that arose in an applied problem, and I would like to know whether it is already in the literature — I suspect it is, but I have not found it. Setup. Let $X_1,\dots,X_K$ be independent homogeneous Poisson processes on $(0,\infty)$ , where $X_b$ has rate $\lambda_b = \kappa c_b$ with $c_b>0$ known and $\kappa>0$ the single unknown parameter. Process $b$ is observed on a window $(0,W_b]$ and every event in that window is recorded. The unusual feature: $W_b$ is not recorded. We know that observation was exhaustive on some window, but not where it ended. What we observe for each $b$ is the number of events $N_b$ and the position $D_b$ of the last one. The two candidate estimators. Write $M=\sum_b N_b$ and $S=\sum_b c_b D_b$ . If one mistakenly treats the data as failure-truncated (observe until the $N_b$ -th event, $N_b$ fixed by design), then $\kappa S\sim\Gamma(M,1)$ exactly and $\tilde\kappa=(M-1)/S$ is unbiased. Under the actual scheme, let $\delta_b=W_b-D_b$ be the unobserved gap between the last event and the end of the window. By memorylessness $\delta_b$ is $\mathrm{Exp}(\lambda_b)$ truncated at $W_b$ , so $\mathbb{E}[\delta_b]=(1-e^{-\lambda_b W_b})/\lambda_b$ . Total exposure is $\sum_b c_b W_b = S+\sum_b c_b\delta_b$ , and since $\mathbb{E}[M]=\kappa\sum_b c_b W_b$ , moment matching gives $$\hat\kappa=\frac{M-K}{S}$$ up to a remainder $\sum_b e^{-\lambda_b W_b}$ , which is negligible whenever each window contains several events. What strikes…