ATM: An aggregation test of moments approach for assessing high‐dimensional normality

Hengjian Cui and Lingyue Zhang

Scandinavian Journal of Statistics, 2026, vol. 53, issue 1, 532-553

Abstract: The Gaussian assumption is the most common and widely used distribution in statistical methodology. Various affine invariant tests for normality rely on the inverse of the covariance matrix and do not apply to high‐dimensional data. The Gaussian moments are equal to the specific values, which implies an exclusive quantitative relationship. This paper introduces two novel indices, the co‐third moment and the co‐fourth moment, to characterize the shape of the distribution relative to the high‐dimensional Gaussian family. Using these indices, two new tests with asymptotic properties under mild regularity conditions defining the subfamily for which tests are theoretically valid are proposed, which substantially avoid using the inverse covariance matrix and are easy to implement. By aggregating the strengths of the two tests and using the power enhancement technique, this study develops a more sensitive test of high‐dimensional normality. Numerical studies and real data analyses provide evidence that the proposed methods achieve well‐controlled size and competitive power, and illustrate their ability to detect departures from Gaussianity across different dimensional settings. The practical steps for empirically checking the regularity conditions are also discussed.

Date: 2026
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