A test for normality based on self-similarity

TL;DR

This paper introduces the Self-Similarity Test for Normality (SSTN), a new statistical test exploiting the self-similarity property of the normal distribution to assess normality in data.

Contribution

The paper proposes a novel normality test based on the self-similarity of the normal distribution, using characteristic functions and Monte Carlo calibration for improved performance.

Findings

SSTN performs competitively with existing tests for normality.

SSTN often outperforms several well-established normality tests.

The test is effective for both small and large samples.

Abstract

Testing for normality is a widely used procedure in statistics and data analysis, often applied prior to employing methods that rely on the assumption of normally distributed data. While several existing tests target distributional characteristics such as higher-order moments, others focus on functional aspects such as the distribution function. In this article, we propose an alternative idea by exploiting the self-similarity property of the normal distribution and introduce the Self-Similarity Test for Normality (SSTN). This procedure leverages the structural property that the distribution of a suitably centered and scaled sum of independent and identically distributed random variables with finite variance coincides with the original distribution if and only if that distribution is normal. The SSTN evaluates normality by applying a self-similarity transformation to the standardized empirical characteristic function and examining how the transformed functions change across successive applications. For the normal distribution, repeated applications preserve the functional form of the characteristic function, whereas deviations from normality manifest in systematic changes between consecutive transforms. These changes are aggregated into a test statistic, whose null distribution is obtained by Monte Carlo calibration, using a sample-size-specific calibration for small samples and an approximation of the asymptotic null distribution for larger ones. A comprehensive simulation study shows that the SSTN performs at least competitively and frequently superior to several well-established tests for normality.