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.

翻译：正态性检验是统计学和数据分析中广泛使用的程序，通常在使用依赖正态分布数据假设的方法之前进行。虽然一些现有检验针对分布特征（如高阶矩），其他检验则关注分布函数等函数方面。本文提出了一种替代思路，利用正态分布的自相似性属性，并引入了自相似性正态性检验（SSTN）。该方法利用了如下结构性质：若且仅当分布为正态时，适当中心化和缩放后的独立同分布（具有有限方差）随机变量之和的分布与其原始分布一致。SSTN通过将自相似性变换应用于标准化经验特征函数，并考察变换函数在连续应用中的变化来评估正态性。对于正态分布，重复应用能保持特征函数的函数形式；而偏离正态性则表现为连续变换之间的系统性变化。这些变化被汇总为一个检验统计量，其零分布通过蒙特卡洛校准获得——小样本时采用样本量特定的校准，大样本时则使用渐近零分布的近似值。全面的模拟研究表明，SSTN在与多种经典正态性检验的比较中至少具有竞争力，且通常表现更优。