The "Straightforward" Nature of Arrival Rate Estimation?

How one estimates the parameter in a Poisson process depends critically on the rule used to terminate the sampling period. For observation until the kth arrival, or observation until time t, well-known maximum likelihood estimators (MLEs) can be used, although they can be biased if the sampling period is such that the expected number of arrivals is small. If one uses a stopping rule such as "observe until the kth arrival or time t," the form of the MLE becomes more complex. In the latter case, it appears a simple ad hoc estimator outperforms its MLE competitor.