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 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].

A Pólya Approximation to the Poisson-Binomial Law

Using Stein's method, we derive explicit upper bounds on the total variation distance between a Poisson-binomial law (the distribution of a sum of independent but not necessarily identically distributed Bernoulli random variables) and a Pólya distribution with the same support, mean, and variance; a nonuniform bound on the pointwise distance between the probability mass functions is also given. A numerical comparison of alternative distributional approximations on a somewhat representative collection of case studies is also exhibited. The evidence proves that no single one is uniformly most accurate, though it suggests that the Pólya approximation might be preferred in several parameter domains encountered in practice.

A Pólya Approximation to the Poisson-Binomial Law

Using Stein's method, we derive explicit upper bounds on the total variation distance between a Poisson-binomial law (the distribution of a sum of independent but not necessarily identically distributed Bernoulli random variables) and a Pólya distribution with the same support, mean, and variance; a nonuniform bound on the pointwise distance between the probability mass functions is also given. A numerical comparison of alternative distributional approximations on a somewhat representative collection of case studies is also exhibited. The evidence proves that no single one is uniformly most accurate, though it suggests that the Pólya approximation might be preferred in several parameter domains encountered in practice.