A Correlation-Free Test for High-Dimensional Elliptical Distributions
arXiv:2607.11304
The paper proposes a test for assessing whether high‑dimensional data follow an elliptical distribution that does not rely on estimating the inverse covariance matrix, and provides theoretical guarantees and bootstrap implementation.
Abstract
Elliptical distributions provide a flexible and widely used extension of multivariate normal distribution. They play a critical role in many statistical procedures when dealing with high-dimensional data. However, goodness-of-fit testing for elliptical distributions remains challenging when the dimension is comparable to or larger than the sample size. In this work, we propose a correlation-free test for high-dimensional elliptical distributions. We establish high-dimensional Gaussian approximation for the test statistic under general correlation structures, allowing the dimension to grow as under finite moment conditions, without using the inverse sample covariance matrix. We further develop Gaussian multiplier bootstrap test procedure and prove its theoretical validity. Numerical studies demonstrate stable finite-sample behavior and favorable power against a range of alternatives. Applications to real datasets illustrate practical utility of the proposed test.