The multivariate generalised Gaussian distribution (MGGD) is commonly used to model high-dimensional vectors with non-Gaussian radial behaviour, ranging from sharp-peaked to heavy-tailed profiles. However, because many classical multivariate tests are based on covariance inversion or high-dimensional density estimation, formal goodness-of-fit assessment for MGGD models remains challenging in modern regimes where the dimension is comparable to or exceeds the sample size. We introduce an affine-invariant, fully non-parametric goodness-of-fit procedure based on the nearest neighbour (NN) graph topology and the adapted zero principle. Following robust standardisation, we construct an independent reference sample from the adapted standardised MGGD and measure, on the combined NN graph, the cross-edge count to assess how well the observed and reference point clouds exhibit the mixture behaviour anticipated by the model. Calibration performed using a refitted parametric bootstrap accounts for nuisance-parameter uncertainty, thus ensuring reliable size under a composite specification. In this paper, we establish asymptotic validity under high-dimensional scaling and demonstrate consistency with respect to fixed elliptical departures, providing a geometric interpretation based on radial concentration and shell separation. Our simulation studies across a broad spectrum of dimensions and tail shapes reveal accurate Type I error control and robust power relative to heavy- and light-tailed alternatives, thus improving upon energy-distance benchmarks and normality-oriented graphical tests in contexts where MGGD modelling is most applicable.

翻译：多元广义高斯分布（MGGD）常用于建模具有非高斯径向行为的高维向量，涵盖从尖峰分布到重尾分布等特征。然而，由于许多经典多元检验依赖于协方差求逆或高维密度估计，在维度与样本量相当或超过样本量的现代场景中，对MGGD模型进行形式化拟合优度评估仍具挑战性。本文提出一种基于最近邻（NN）图拓扑结构与自适应零原则的仿射不变全非参数拟合优度检验方法。经稳健标准化后，我们从自适应标准MGGD中构建独立参考样本，并在组合NN图上度量跨边计数，以评估观测点云与参考点云是否展示出模型预期的混合行为。采用基于重拟合参数自助法的校准过程可消除冗余参数不确定性，从而确保复合设定下的可靠检验尺度。本文建立了高维渐进有效性理论，并在固定椭圆偏离假设下证明一致性，同时基于径向浓度与壳分离提供几何解释。针对不同维度和尾部形态的广泛模拟研究显示，该方法在控制第一类错误率方面表现精确，且对重尾与轻尾备择假设均具有稳健检验功效——相较于能量距离基准和正态性导向图检验，在MGGD建模适用场景中优势显著。