# Estimate partially observed Poisson process

I try to estimate the intensity of a Poisson process $$P_1$$, but it is not fully observable. There are some "obervers" coming to the system which follow another Poisson process $$P_2$$. In reality, $$P_1$$ and $$P_2$$ cannot come at the same time, but we can take them as independent for simplicity. $$P_2$$ itself is fully observable, so we can estimate its intensity.

At every periodic time window $$\Delta$$ (known), when the observer comes before time $$\Delta_o$$ ($$\Delta_o$$ < $$\Delta$$) the number of arrivals in $$P_1$$ can be observed. So, there are some duration that $$P_1$$ is not observable. In reality, $$\Delta_o$$ also depends on the number of arrival in $$P_1$$, but we can take it as a known fixed value. The observation interval follows an explantional distribution because of the Poisson process $$P_2$$.

The first observation interval is $$[0 + c \Delta, t_{21}^c + c \Delta], c = 0,1, \cdots$$, and the latter is $$[t_{2,i-1}^c + c \Delta, t_{2,i}^c + c \Delta]$$ if $$t_{2,i}^c < \Delta_o$$, where $$t_{2,i}^c$$ is the arrival time of $$i$$th observer in cycle $$c$$.

Is the observation of $$P_1$$ a compound Poisson process? Is there any research on this kind of problem? If the Poisson process $$P_1$$ and $$P_2$$ are nonhomogeneous Poisson process, how to estimate the intensity of them?

• Can you specify more clearly what you mean by periodic time window? Do you mean that a new observation interval starts at times $t = k\Delta$, $n=0,1,\ldots$ and ends some time later at $t=k\Delta + T_k$ where the $T_k>0$ are random variables of which the distribution depends on a Poisson process $P_2$? How exactly do the $T_k$ depend on $P_2$? Are the Poisson processes $P_1$ and $P_2$ independent? – StijnDeVuyst Oct 24 at 17:42
• @StijnDeVuyst I modified the question description, could you take a look at it? – WZhao Oct 24 at 19:03