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

Fraser Daly

TL;DR

This work studies, for iid samples, the distributional error in approximating the number of observations attaining the sample maximum (discrete case) or within a threshold of a sample order statistic (continuous case). It develops explicit total variation bounds using Stein's method, introducing a novel logarithmic target distribution for the discrete setting and applying negative binomial approximations via mixed binomial representations for the continuous setting. The authors provide concrete bounds and illustrative examples (geometric, Gumbel, Uniform) and establish a Poisson approximation in the discrete case as a complementary tool. The results yield practical, quantitative error guarantees for simple, well-known distributions (logarithmic, NB, Poisson) as approximants to complex maxima-related statistics, with potential extensions to dependent data and broader applications in reliability and algorithmic contexts.

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.

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

TL;DR

This work studies, for iid samples, the distributional error in approximating the number of observations attaining the sample maximum (discrete case) or within a threshold of a sample order statistic (continuous case). It develops explicit total variation bounds using Stein's method, introducing a novel logarithmic target distribution for the discrete setting and applying negative binomial approximations via mixed binomial representations for the continuous setting. The authors provide concrete bounds and illustrative examples (geometric, Gumbel, Uniform) and establish a Poisson approximation in the discrete case as a complementary tool. The results yield practical, quantitative error guarantees for simple, well-known distributions (logarithmic, NB, Poisson) as approximants to complex maxima-related statistics, with potential extensions to dependent data and broader applications in reliability and algorithmic contexts.

Abstract

In the setting where we have independent observations of a random variable , 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 is discrete) or the number of observations within a given distance of an order statistic of the sample (in the case where 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 is geometric, Gumbel and uniform are used as illustrative examples.
Paper Structure (11 sections, 6 theorems, 77 equations, 2 figures, 1 table)

This paper contains 11 sections, 6 theorems, 77 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. The discrete case
  3. The absolutely continuous case
  4. Stein's method for the logarithmic distribution
  5. Logarithmic approximation for the negative binomial distribution
  6. Proof of Theorem \ref{['thm:discrete']}(a)
  7. Negative binomial approximation for mixed binomials
  8. Proof of Theorem \ref{['thm:discrete']}(b)
  9. Proof of Theorem \ref{['thm:continuous']}
  10. Poisson approximation for $K_n$ in the discrete case
  11. Concluding remarks

Key Result

Theorem 1

Let $X_1,\ldots,X_n$ be positive, integer-valued, i$.$i$.$d$.$ random variables with maximum $M_n$. Let $K_n$ be defined as in eq:kdef, with probability mass function as in eq:kmass.

Figures (2)

  • Figure 1: The upper bound of Theorem \ref{['thm:discrete']}(a) in the case where $X$ has a geometric distribution, evaluated for $n=20$ and various values of $p$
  • Figure 2: The upper bound \ref{['eq:gumbel']} for the case where $X$ has a Gumbel distribution, evaluated for various values of $a$ in the cases $n=20$ and $n=100$

Theorems & Definitions (15)

  • Theorem 1
  • Example 2
  • Theorem 3
  • Theorem 5
  • Example 6
  • Example 7
  • Example 8
  • Theorem 9
  • proof
  • Remark 10
  • ...and 5 more