Abstract

We propose and study a scale-invariant kurtosis-type statistic for testing Gaussianity in the setting of complex matrix-variate scale mixtures and show it becomes pivotal with respect to the mixing distribution, enabling tractable Gaussianity testing without explicit integration over the mixing law.
Let $X \in \mathbb{C}^{n \times p}$ and consider the quadratic form $S = X^H A X$, where $A$ is a fixed Hermitian positive definite matrix.
We consider the invariant statistic $T = \mathrm{tr}(S^2)/(\mathrm{tr}(S))^2$, and derive its expectation and variance under the complex Gaussian model.
Interestingly, $T$ is invariant to multiplicative scale mixtures, eliminating dependence on latent scaling variables while retaining sensitivity to deviations from Gaussian structure through higher-order spectral dispersion.
This allows for a computationally efficient and theoretically tractable test.
We further characterize its behavior under general scale mixtures and demonstrate its effectiveness in detecting heavy-tailed alternatives.
Simulation studies indicate strong discriminatory power between (complex) Gaussian and heavy-tailed models.
Extensions to structured quadratic forms and signal detection settings are discussed.