= Poisson observation scheme and rate likelihood
{title2=$\widehat\lambda=N_H/H\quad\text{or}\quad n/T_n$}
A <Poisson process> observed for a fixed duration $H$ has likelihood proportional to $\lambda^{N_H}e^{-\lambda H}$. Complete independent <exponential distribution> gaps observed until a fixed event count $n$ have likelihood $\lambda^ne^{-\lambda T_n}$. Both give count divided by exposure, but the exposure is random in the second design. Under fixed exposure the total count is sufficient: conditional on it, the event locations are independent of the rate, so exact timestamps add no rate information to interval counts. Gap estimates and fixed-horizon estimates should not be compared without distinguishing their stopping rules and any final censored gap.
Back to article page
