I have a Poisson process where the times between events are exponential random variables with mean, $\mu$. I also have an interval of time defined by a start time, $s$ and an end time, $e$. Now, I want to estimate $\mu$. One approach is to take the events that fall inside the time window and average their durations.
For instance, in the figure below, the crosses represent the events from the Poisson process and the two event durations shaded blue are averaged.
Simulations seem to suggest this is a biased estimator for $\mu$. Trying to find the reason this might be the case.
The estimator is biased because it doesn't correctly reflect the arrivals that occur in $[s,e]$. Let $\tau = \sup\{n\geqslant0: T_n < s\}$ - this is the time of the first arrival in a given sample. So you're really counting arrivals in $[\tau,e]$, as the backward recurrence time in a Poisson process is exponential with the same rate, due to the memoryless property.
The probability that there is an arrival in $[0,s)$ is $$ \mathbb P(N(s)>0) = 1-e^{-\frac1\mu s}, $$ and hence with this probability your time window will be longer than $e-s$.