Conventional methods for testing independence between two Gaussian vectors require sample sizes greater than the number of variables in each vector. Therefore, adjustments are needed for the high-dimensional situation, where the sample size is smaller than the number of variables in at least one of the compared vectors. It is critical to emphasize that the methods available in the literature are unable to control the Type I error probability under the nominal level. This fact is evidenced through an intensive simulation study presented in this paper. To cover this lack, we introduce a valid randomized test based on the Kronecker delta covariance matrices estimator. As an empirical application, based on a sample of companies listed on the stock exchange of Brazil, we test the independence between returns of stocks of different sectors in the COVID-19 pandemic context.

测试两个高斯向量之间独立性的传统方法需要的样本量大于每个向量中的变量数。因此，需要针对高维情况进行调整，其中样本量小于至少一个比较向量中的变量数。需要强调的是，文献中可用的方法无法将 I 类错误概率控制在标称水平以下。这一事实通过本文提出的深入模拟研究得到证实。为了弥补这一不足，我们引入了基于 Kronecker delta 协方差矩阵估计器的有效随机测试。作为实证应用，基于巴西证券交易所上市公司的样本，