Due to the broad applications of elliptical models, there is a long line of research on goodness-of-fit tests for empirically validating them. However, the existing literature on this topic is generally confined to low-dimensional settings, and to the best of our knowledge, there are no established goodness-of-fit tests for elliptical models that are supported by theoretical guarantees in high dimensions. In this paper, we propose a new goodness-of-fit test for this problem, and our main result shows that the test is asymptotically valid when the dimension and sample size diverge proportionally. Remarkably, it also turns out that the asymptotic validity of the test requires no assumptions on the population covariance matrix. With regard to numerical performance, we confirm that the empirical level of the test is close to the nominal level across a range of conditions, and that the test is able to reliably detect non-elliptical distributions. Moreover, when the proposed test is specialized to the problem of testing normality in high dimensions, we show that it compares favorably with a state-of-the-art method, and hence, this way of using the proposed test is of independent interest.

翻译：由于椭圆模型具有广泛的应用，长期以来一直存在一系列关于经验验证其拟合优度的检验研究。然而，该主题的现有文献通常局限于低维设置，且据我们所知，目前尚无在高维情况下具有理论保证的、成熟的椭圆模型拟合优度检验方法。本文针对此问题提出了一种新的拟合优度检验方法，我们的主要结果表明，当维度与样本量成比例发散时，该检验是渐近有效的。值得注意的是，该检验的渐近有效性并不需要对总体协方差矩阵做出任何假设。在数值性能方面，我们证实该检验的经验水平在一系列条件下均接近名义水平，并且能够可靠地检测非椭圆分布。此外，当将所提出的检验专门用于高维正态性检验问题时，我们证明其性能优于现有先进方法，因此，这种使用所提出检验的方式具有独立的研究价值。