It is assumed that we observe one realization of an $r$-dimensional counting process with intensities that are products of a predictable weight process, a common function of time and parameters $\beta_i, i = 1, \cdots, r$, which distinguish the components. Provided the realization observed brings increasing information on $\beta$ as the observed time grows, strong consistency of a partial ML estimator is proved. For such realizations it is also proved that the estimate, after applying a random normalization, is asymptotically standard normal.
"Estimation in some Counting Process Models with Multiplicative Structure." Ann. Statist. 17 (4) 1501 - 1508, December, 1989. https://doi.org/10.1214/aos/1176347378