# Can I estimate the parameter of a Poisson arrival process from a low-incidence observation period?

If I know only that the arrival process is Poisson, and I observe it for a pre-chosen (say, unit) period of time, observing $k$ arrivals, is it meaningful to describe an estimate of its time parameter from the observation as (A) maximum-likelihood, (B) minimum-variance, or (C) unbiased if $k$ is very small? If $k$ is very large, then something like $1/k$ would be a pretty plausible estimate by any criterion. For moderate values of $k$, I guess that the answer is still positive, although more complicated. But I am interested in the case when $k=2$ or even $k=1$.

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If you estimate the arrival rate as $k$/unit time, the estimate is indeed A) maximum likelihood, B) minimum variance, and C) unbiased, regardless of sample size. –  jbowman Jun 26 '12 at 18:25
Do you only need a point estimate, or did you want an interval? –  Glen_b Jun 27 '12 at 8:53
@jbowman:From your comment, the corresponding estimate for the time parameter, $1/k$, is maximum-likelihood too. But is it also minimum-variance and unbiased? –  John Bentin Jun 27 '12 at 22:21
@Glen_b: I'm only looking for a point estimate. –  John Bentin Jun 27 '12 at 22:23

Unfortunately, the maximum likelihood estimate of the rate parameter for a Poisson process that's sampled over a predetermined interval $T$ does not have a finite mean (or higher moments). This is because there's a nonzero probability of seeing no arrivals ($k=0$) in $T$, leading to an estimate of $T/0$. We can fix that problem in any number of ways, an obvious one being to use $\max\{k, c\}$ instead of $k$ where $c$ is some constant such as 1/2. The estimate is no longer maximum likelihood, nor is it unbiased, but at least it has moments of all order, and it is consistent.

Also unfortunately, we can't really describe an estimator as "unbiased given $k=1$", as after we've observed the data, there's no randomness left (since $T$ is fixed also.)

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Do you really need the maximum likelihood estimate? Can you get away with the Bayesian estimator of the intensity of the Poisson process? For $k$ observed events in a period of length $T$, the likelihood over the intensity is gamma-distributed with shape $k$ and rate $T$. This is correct even if $k$ is one or zero (although you might want to introduce a gamma-distributed prior).