Edgeworth expansion for Bernoulli weighted mean
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In this work, we derive an Edgeworth expansion for the Bernoulli weighted mean μ̂ = ∑_i=1^n Y_i T_i/∑_i=1^n T_i in the case where Y_1, …, Y_n are i.i.d. non semi-lattice random variables and T_1, …, 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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