A Correlation-Free Test for High-Dimensional Elliptical Distributions

2026-07-13

AI Summary

This study addresses the challenging problem of goodness-of-fit testing for high-dimensional elliptical distributions, particularly when the dimensionality is comparable to or exceeds the sample size and complex correlation structures are present. The authors propose a novel test statistic that circumvents the need to estimate the inverse of the covariance matrix, thereby avoiding instability in high dimensions. Under only finite moment conditions, they establish a Gaussian approximation for the proposed statistic and develop a theoretically justified Gaussian multiplier bootstrap procedure—the first of its kind for high-dimensional ellipticity testing. Notably, their method accommodates dimensions satisfying $\log p = o(n^{1/14})$. Numerical experiments demonstrate robust finite-sample performance and high power against various alternatives, while real-data analyses confirm its practical feasibility and effectiveness.

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 $\log p=o(n^{1/14})$ 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.

Problem

• high-dimensional
• elliptical distributions
• goodness-of-fit testing
• correlation-free

Innovation

• correlation-free test
• high-dimensional elliptical distributions
• Gaussian approximation
• Gaussian multiplier bootstrap
• goodness-of-fit testing