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Elementary Tail Bounds on the Hypergeometric Distribution

Vaisakh Mannalath, Víctor Zapatero, Marcos Curty

TL;DR

The paper addresses tight tail bounds for the Hypergeometric distribution under finite-population sampling. It introduces an elementary approach that leverages a permutation symmetry between the sample size n and the number of successes K to produce Chernoff-type bounds that reflect the sampling fraction n/N, and connects to the Poisson-binomial representation and Hoeffding bounds to derive a beta-type bound. It also provides a factorial-moment bound that consistently outperforms standard Chernoff in the tested regimes. Empirical results and discussion show that symmetry-based bounds and the factorial-moment bound yield tighter estimates, particularly when the sampling fraction is large or when n > K, enhancing practical uncertainty quantification for sampling without replacement.

Abstract

We use a simple method to derive two concentration bounds on the hypergeometric distribution. Comparison with existing results illustrates the advantage of these bounds across different regimes.

Elementary Tail Bounds on the Hypergeometric Distribution

TL;DR

The paper addresses tight tail bounds for the Hypergeometric distribution under finite-population sampling. It introduces an elementary approach that leverages a permutation symmetry between the sample size n and the number of successes K to produce Chernoff-type bounds that reflect the sampling fraction n/N, and connects to the Poisson-binomial representation and Hoeffding bounds to derive a beta-type bound. It also provides a factorial-moment bound that consistently outperforms standard Chernoff in the tested regimes. Empirical results and discussion show that symmetry-based bounds and the factorial-moment bound yield tighter estimates, particularly when the sampling fraction is large or when n > K, enhancing practical uncertainty quantification for sampling without replacement.

Abstract

We use a simple method to derive two concentration bounds on the hypergeometric distribution. Comparison with existing results illustrates the advantage of these bounds across different regimes.
Paper Structure (2 sections, 2 theorems, 17 equations, 2 figures)

This paper contains 2 sections, 2 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results

Key Result

Proposition 1

Let $X \sim \text{Hypergeometric}(N, K, n)$. Then, and for all integer $d\geq{}nK/N+1$.

Figures (2)

  • Figure 1: Upper tail probability, $\Pr[X\geq n(K/N +\delta)]$, as a function of the relative deviation $\delta$ with respect to the mean, for $X\sim \text{Hypergeometric}(N,K,n)$ and a population size $N = 1000$. Left column: $K/N = 2\%$. Right column: $K/N = 5\%$. Blue and yellow lines: Chernoff and $\beta$ bounds with (solid) and without (dashed) invoking the permutation symmetry of $n$ and $K$. Green line: best-performing bound among Serfling, Hush, Chatterjee, Goldstein, Rohde, Bardenet and Theorem 1 and 3 of Greene. Purple line: Factorial moment bound. Red line: exact cumulative mass function of $X$.
  • Figure 2: $\Pr[X\geq n(K/N +\delta)]$ as a function of $\delta$, for $X\sim \text{Hypergeometric}(N,K,n)$ and $N = 10000$. The followed criteria are common with those of Figure \ref{['fig:upper_tail_1000']}.

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2