The median of a jittered Poisson distribution
Let $N_λ$ and $U$ be two independent random variables respectively distributed as a Poisson distribution with parameter $λ>0$ and a uniform distribution on $(0,1)$. This paper establishes that the median, say $M$, of $N_λ+U$ is close to $λ+1/3$ and more precisely that $M-λ-1/3=o(λ^{-1})$ as $λ\to \infty$. This result is used to construt a very simple robust estimator of $λ$ which is consistent and asymptotically normal. Compared to known robust estimates, this one can still be used with large datasets ($n\simeq 10^9$).
math.ST↗