Parametric confidence regions for means of symmetric positive matrices



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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 (2015) introduced a lognormal distribution on the space of SPD matrices, providing a convenient framework for parametric inference on this space. In the first part of this dissertation our goal is to check how robust confidence regions based on this distributional assumption are to a lack of lognormality, and in the second part to introduce a Cholesky normal distribution for SPD matrices and define new parametric confidence regions for the mean of SPD matrices. The methods in the first part are illustrated in a simulation study by examining the coverage probability of various mixtures of lognormal distributions. The methods in the second part are also illustrated in simulations study by investigating the properties of the Cholesky normal distribution and by developing new confidences regions based on this distribution via coverage probability.