# The median of a jittered Poisson distribution

## TL;DR

This paper analyzes the median of a jittered Poisson distribution, showing it approximates λ+1/3 for large λ, and introduces a simple, robust estimator for λ that is effective even with very large datasets.

## Contribution

It establishes the asymptotic behavior of the median of a jittered Poisson distribution and proposes a new robust estimator for the Poisson parameter λ.

## Key findings

- Median of N_λ + U is close to λ + 1/3 as λ→∞
- Proposed estimator is consistent and asymptotically normal
- Estimator is suitable for large datasets (up to 10^9 observations)

## Abstract

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 construt 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 ($n\simeq 10^9$).