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Large Deviations Inequalities for Unequal Probability Sampling Without Replacement

Dean P. Foster, Sergiu Hart

Abstract

We provide bounds on the tail probabilities for simple procedures that generate random samples _without replacement_, when the probabilities of being selected need not be equal.

Large Deviations Inequalities for Unequal Probability Sampling Without Replacement

Abstract

We provide bounds on the tail probabilities for simple procedures that generate random samples _without replacement_, when the probabilities of being selected need not be equal.

Paper Structure

This paper contains 5 sections, 2 theorems, 24 equations.

Table of Contents

  1. Martingale-Based Procedures
  2. General Procedures $\mathfrak{X}$
  3. “ In Order" Procedures $\mathfrak{X}^{\ast }$
  4. Using the Bounds
  5. Appendix. Tail Probability Bounds

Key Result

Theorem 1

The procedures $\mathfrak{X}$ satisfy $\mathbb{P}\left[ i\in S\right] =kw^{i}$ for every $i$ in $[n]$, and for every set $A\subset \lbrack n]$ and $\delta \geq 0$, where is the relative weight of $A$, and

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2