Testing the hypothesis of a nested block covariance matrix structure with applications to medicine and natural sciences

This paper addresses the challenge of testing the hypothesis of what the authors call a nested block circular‐compound symmetric (NBCCS) covariance structure for the population covariance matrix. This is a covariance structure which has an outer block compound symmetric structure, where the diagonal blocks are themselves block circular matrices, while the off‐diagonal blocks are formed by all equal matrices. The NBCCS null hypothesis is decomposed into sub‐hypotheses, allowing this way for a simpler way to obtain a likelihood ratio test and its associated statistic. The exact moments of this statistic are derived, and its distribution is carefully examined. Given the complicated nature of this distribution, highly precise near‐exact distributions were developed. Numerical studies are conducted to assess the proximity between these near‐exact distributions and the exact distribution, highlighting the performance of these approximations, even in the case of very small sample sizes. Furthermore, three datasets, on bone mineral content, metabolic rates of glucose, and soil moisture are used to exemplify the practical application of the methodology derived in this study.