The median of a jittered Poisson distribution
Résumé
Let $N-\lambda$ and $U$ be two independent random variables respectively distributed as a Poisson distribution with parameter $\lambda>0$ and a uniform distribution on $(0, 1)$. This paper establishes that the median, say $M$, of $N_\lambda+U$ is close to $\lambda+1/3$ and more precisely that $M−\lambda−1/3=o(\lambda^{−1})$ as $\lambda\to\infty$. This result is used to construct a very simple robust estimator of $\lambda$ which is consistent and asymptotically normal. Compared to known robust estimates, this one can still be used with large datasets ($n\simeq10^9$).