A Nonparametric Goodness-of-Fit Test for High-Dimensional Generalized Gaussian Distributions via Nearest-Neighbor Graphs

Abstract

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.