Zero biasing in one and higher dimensions, and applications

Given any mean zero, finite variance σ² random variable W, there exists a unique distribution on a variable W* such that EW f(W) = σ²Ef′(W*) for all absolutely continuous functions f for which these expectations exist. This distributional ‘zero bias’ transformation of W to W*, of which the normal is the unique fixed point, was introduced in Goldstein & Reinert (1997) to obtain bounds in normal approximations. After providing some background on the zero bias transformation in one dimension, we extend its definition to higher dimension and illustrate with an application to the normal approximation of sums of vectors obtained by simple random sampling.