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?

  • $\begingroup$ 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? $\endgroup$ – StijnDeVuyst Oct 24 at 17:42
  • $\begingroup$ @StijnDeVuyst I modified the question description, could you take a look at it? $\endgroup$ – WZhao Oct 24 at 19:03

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