Edgeworth expansion for Bernoulli weighted mean

Abstract

In this work, we derive an Edgeworth expansion for the Bernoulli weighted mean $\hat{\mu }=\frac{{\sum }_{i=1}^n{Y}_i{T}_i}{{\sum }_{i=1}^n{T}_i}$ in the case where $Y_1,\dots ,Y_n$ are i.i.d. non semi-lattice random variables and $T_1,\dots ,T_n$ 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.

12 pages