# Estimation of right truncated poisson process

I have following problem:

Imagine I generate large number of homogenous poisson process sample paths (by sample path I mean a sequence of arrival times $\tau_i$ all with the same intensity. However these paths are generated on relatively short interval [0,T] so for most time I observe a realization with no arrival.

Now I would like to estimate intensity of this process. My idea was to use likelihood function conditioned on number of arrival times >=1, however in this case I would effectively dispose all sample paths with no arrival observed and even for this case I am not sure how to do that.

Any relevant literature is welcome!

• if you have $n$ independent periods of length $T$ to estimate the parameters of the process, isn't this effectively equivalent to having $1$ period of length $nT$? Does that work to construct hypothetical data as if all the periods were consecutive? – Matthew Gunn Dec 11 '17 at 23:08
• Hey Mathew! I suppose that this could work for homogenous poisson, however I want to advance this procedure to self-exciting process later on for which this would not hold. Hence keeping the estimation procedure as outlined is essential for me. But again if you have any good link on estimation of truncated processes please share it with me. – Michael Mark Dec 12 '17 at 9:02