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The median of a jittered Poisson distribution

Abstract

Let NλN_\lambda and UU be two independent random variables respectively distributed as a Poisson distribution with parameter λ>0\lambda >0 and a uniform distribution on (0,1)(0,1). This paper establishes that the median, say MM, of Nλ+UN_\lambda+U is close to λ+1/3\lambda +1/3 and more precisely that Mλ1/3=o(λ1)M-\lambda-1/3=o(\lambda^{-1}) as λ\lambda\to \infty. This result is used to construt a very simple robust estimator of λ\lambda which is consistent and asymptotically normal. Compared to known robust estimates, this one can still be used with large datasets (n109n\simeq 10^9)

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