Edgeworth expansion for Bernoulli weighted mean

2022/08/19 by Pierre-Louis Cauvin, Cauvin, Pierre-Louis
Decision Sciences · Mathematics · #Advanced Statistical Process Monitoring #Applied mathematics #Bernoulli distribution #Bernoulli's principle #Combinatorics #Discrete mathematics #Edgeworth series #Equivalence (formal languages) #Generalization #Lattice (music) #Mathematical analysis #Mathematics #Physics #Probability and Risk Models #Random variable #Statistical Distribution Estimation and Applications #Statistical physics #Statistics #Thermodynamics #math.PR #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.2208.09274

published in arXiv (Cornell University) (Cornell University) · 12 pages

arxiv created 2022/08/19 · openalex publication_date 2022/08/19 · arxiv updated 2022/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

In this work, we derive an Edgeworth expansion for the Bernoulli weighted mean μ = \frac∑_i=1ⁿ Y_i T_i∑_i=1ⁿ T_i in the case where Y₁, …, Y_n are i.i.d. non semi-lattice random variables and T₁, …, T_n are Bernoulli distributed random variables with parameter p. We also define the notion of a semi-lattice distribution, which gives a more geometrical equivalence to the classical Cramér's condition in dimensions bigger than 1. Our result provides a first step into the generalization of classical Edgeworth expansion theorems for random vectors that contain both semi-lattice and non semi-lattice variables, in order to prove consistency of bootstrap methods in more realistic setups, for instance in the use case of online AB testing.

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