L1 bounds in normal approximation

Abstract

The zero bias distribution W^* of W, defined though the characterizing equation EW f(W)=σ²E f'(W^*) for all smooth functions f, exists for all W with mean zero and finite variance σ². For W and W^* defined on the same probability space, the L¹ distance between F, the distribution function of W with EW=0 and Var(W)=1, and the cumulative standard normal Φ has the simple upper bound

‖F−Φ‖₁≤2E|W^*−W|.

This inequality is used to provide explicit L¹ bounds with moderate-sized constants for independent sums, projections of cone measure on the sphere S(ℓ_n^p), simple random sampling and combinatorial central limit theorems.