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!