The median of a jittered Poisson distribution

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