# On moderate deviations in Poisson approximation

## TL;DR

This paper investigates the accuracy of Poisson approximation for rare event counts, showing that with proper adjustments, the error estimates for moderate deviations are improved and applicable across various problems.

## Contribution

It introduces refined error estimates for moderate deviations in Poisson approximation, applicable to multiple classical problems, with no unspecified constants and easy to implement.

## Key findings

- Poisson tail probabilities better approximate rare event counts than normal tails.
- Adjusted estimates improve error bounds for moderate deviations.
- Applications include Poisson-binomial, matching, occupancy, birthday, random graphs, and 2-runs.

## Abstract

In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems in six applications: Poisson-binomial distribution, matching problem, occupancy problem, birthday problem, random graphs and 2-runs. The paper complements the works of \cite{CC92,BCC95,CFS}.