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

## TL;DR

This paper introduces two novel goodness-of-fit tests for univariate normality based on distributional fixed point properties, demonstrating their theoretical validity and competitive power through empirical studies.

## Contribution

The paper develops new normality tests using $L^2$-distances related to the Stein characterization, with proven asymptotic properties and practical performance evaluation.

## Key findings

- Tests are theoretically justified under null and alternative hypotheses.
- Empirical critical values enable practical application.
- New tests show competitive power in finite samples.

## 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 normal distribution or the empirical distribution of the data, respectively. Weak convergence results are derived under the null hypothesis, under fixed alternatives as well as under contiguous alternatives. Empirical critical values are provided and a comparative finite-sample power study shows the competitiveness to classical procedures.