Robustness of lognormal confidence regions for means of symmetric positive definite matrices when applied to mixtures of lognormal distributions

Symmetric positive definite (SPD) matrices arise in a wide range of applications including diffusion tensor imaging (DTI), cosmic background radiation, and as covariance matrices. A complication when working with such data is that the space of SPD matrices is a manifold, so traditional statistical methods may not be directly applied. However, there are nonparametric procedures based on resampling for statistical inference for such data, but these can be slow and computationally tedious. Schwartzman (Int Stat Rev 84(3):456–486, 2016). introduced a lognormal distribution on the space of SPD matrices, providing a convenient framework for parametric inference on this space. Our goal is to check how robust confidence regions based on this distributional assumption are to a lack of lognormality. The methods are illustrated in a simulation study by examining the coverage probability of various mixtures of distributions.