Testing normality via a distributional fixed point property in the Stein characterization

Abstract

We propose two families of tests for the classical goodness-of-fit problem to univariate normality. The new procedures are based on $L^2$-distances of the empirical zero-bias transformation to the empirical distribution or the normal distribution function. Weak convergence results are derived under the null hypothesis, under contiguous as well as under fixed alternatives. A comparative finite-sample power study shows the competitiveness to classical procedures.