On moderate deviations in Poisson approximation

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 a Poisson distribution than those of the normal distribution. We then show that the mode...

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Bibliographic Details
Published in:Journal of applied probability 2020-09, Vol.57 (3), p.1005-1027
Main Authors: Liu, Qingwei, Xia, Aihua
Format: Article
Language:English
Subjects:
Approximation
Mathematical analysis
Normal distribution
Occupancy
Poisson distribution
Random variables
Ratios
Research Papers
Statistical analysis
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Summary: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 a Poisson distribution than those of the normal distribution. We then show that 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 [18]. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems via six applications: Poisson-binomial distribution, the matching problem, the occupancy problem, the birthday problem, random graphs, and 2-runs. The paper complements the works [16], [8], and [18].
ISSN:0021-9002
1475-6072
DOI:10.1017/jpr.2020.47