Estimating a Poisson Arrival Rate with Zero-Arrival Samples

Summary

The document poses an estimation problem for a homogeneous Poisson process observed over short intervals. Because many simulated paths contain no arrivals, it asks how to estimate the common intensity while accounting for those zero-count observations. It considers conditioning the likelihood on observing at least one arrival, which would discard paths with no arrivals, but gives no derivation or recommended estimator.

The material is therefore a useful statement of a statistical issue rather than a worked method. It highlights the distinction between using the full count data and estimating from a sample selected for having a positive count. No simulations, numerical evidence, or literature references are provided, and the document does not resolve which approach is appropriate. Its scope is rate estimation for a homogeneous Poisson process over a fixed observation interval.

Key ideas

• Short observation windows can produce many Poisson samples with zero arrivals.
• Conditioning on a positive arrival count excludes zero-count paths and changes the data used for estimation.
• The document raises the intensity-estimation problem but does not provide an estimator or likelihood solution.
• The setup assumes a homogeneous Poisson process observed over a fixed interval.

Tags

# Estimation of right truncated poisson process


# 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!

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