Approximations for the number of maxima and near-maxima in independent data

TL;DR

This paper develops explicit error bounds for approximating the count of sample maxima or near-maxima in independent data, using Stein's method for various distributions.

Contribution

It introduces new approximation techniques with explicit error bounds for maxima counts, including Stein's method for logarithmic and negative binomial distributions.

Findings

Explicit total variation bounds for maxima and near-maxima counts

Stein's method developed for logarithmic and negative binomial distributions

Applications to geometric, Gumbel, and uniform distributions

Abstract

In the setting where we have $n$ independent observations of a random variable $X$, we derive explicit error bounds in total variation distance when approximating the number of observations equal to the maximum of the sample (in the case where $X$ is discrete) or the number of observations within a given distance of an order statistic of the sample (in the case where $X$ is absolutely continuous). The logarithmic and Poisson distributions are used as approximations in the discrete case, with proofs which include the development of Stein's method for a logarithmic target distribution. In the absolutely continuous case our approximations are by the negative binomial distribution, and are established by considering negative binomial approximation for mixed binomials. The cases where $X$ is geometric, Gumbel and uniform are used as illustrative examples.