Conference Agenda

Exact testing of model assumptions is often of limited relevance, especially in high-dimensional settings. Structural assumptions on large-dimensional covariance matrices, such as sphericity, are rarely expected to hold exactly for real data, and practitioners are often primarily interested in whether such model assumptions are approximately satisfied. In this work, we propose a test for approximate sphericity of high-dimensional covariance matrices, where the tolerated level of deviation from sphericity can be chosen by the user. Our test statistic is based on estimators of the largest and smallest eigenvalues of the population covariance matrix in a high-dimensional regime, where the corresponding sample eigenvalues are not consistent. We derive theoretical guarantees showing that the test keeps the prescribed asymptotic level under the null hypothesis and is power consistent under the alternative. Our key theoretical contribution is a joint central limit theorem for the estimators of the extreme eigenvalues of the population covariance matrix, provided the corresponding eigenvalues exceed the critical phase transition threshold.