Reliable Covariance Estimation

Covariance or scatter matrix estimation is ubiquitous in most modern statistical and machine learning applications. The task becomes especially challenging since most real-world datasets are essentially non-Gaussian. The data is often contaminated by outliers and/or has heavy-tailed distribution causing the sample covariance to behave very poorly and calling for robust estimation methodology. The natural framework for the robust scatter matrix estimation is based on elliptical populations. Here, Tyler's estimator stands out by being distribution-free within the elliptical family and easy to compute. The existing works thoroughly study the performance of Tyler's estimator assuming ellipticity but without providing any tools to verify this assumption when the covariance is unknown in advance. We address the following open question: Given the sampled data and having no prior on the data generating process, how to assess the quality of the scatter matrix estimator? In this work we show that this question can be reformulated as an asymptotic uniformity test for certain sequences of exchangeable vectors on the unit sphere. We develop a consistent and easily applicable goodness-of-fit test against all alternatives to ellipticity when the scatter matrix is unknown. The findings are supported by numerical simulations demonstrating the power of the suggest technique.