The efficiencies of the spatial median and spatial sign covariance matrix for elliptically symmetric distributions

The spatial median and spatial sign covariance matrix (SSCM) are popularly used robust alternatives for estimating the location vector and scatter matrix when outliers are present or it is believed the data arises from some distribution that is not multivariate normal. When the underlying distribution is an elliptical distribution, it has been observed that these estimators perform better under certain scatter structures. This dissertation is a detailed study of the efficiencies of the spatial median and the SSCM under the elliptical model, in particular the dependence of their efficiencies on the population scatter matrix. For the spatial median, it is shown this estimator is asymptotically most efficient compared to the MLE for the location vector when the population scatter matrix is proportional to the identity matrix. Furthermore, it is possible to construct an affinely equivariant version of the spatial median that is asymptotically more efficient than the spatial median. Asymptotic relative efficiencies of these two estimators are calculated to demonstrate how inefficient the spatial median can be as the underlying scatter structure becomes more elliptical. A simulation experiment is carried out to provide evidence of analogous result for finite samples. When the goal is estimating eigenprojection matrices, it is proven that under the elliptical model the eigenprojection estimates obtained from the Tyler matrix are asymptotically more efficient than those corresponding to the SSCM. Calculations of asymptotic relative efficiencies are presented to demonstrate the loss of efficiency in using eigenprojection estimates of the SSCM as opposed to the Tyler matrix, particularly when the scatter structure of the data is far from spherical. To assess the performance of these estimators in the finite sample setting, the notion of principal angles is used to define a means to compare eigenprojection estimators. Using this concept, simulations are implemented that support finite sample results similar to those for the asymptotic case. The implications of the above results are discussed, particularly in the application of principal component analysis. Future research directions are then proposed.Ph. D.Includes bibliographical referencesIncludes vitaby Andrew F. Magya