On estimating the eigenvalues of a covariance matrix.

Three methods for estimating the eigenvalues of the parameter covariance matrix in a Wishart distribution are investigated. Since the sample eigenvalues are biased, these procedures either shrink the sample estimates towards some central value or derive estimates that follow an estimated model for the eigenstructure. First, a method due to Dey of shrinking the sample eigenvalues towards their geometric mean is considered. An improvement to Dey's estimator is given that does not increase the bias of the smallest eigenvalue estimates. It is shown that under certain conditions this improved estimator dominates both Dey's estimator and the eigenvalues of the sample covariance matrix under a squared error loss. Second, improved estimators of the elementary symmetric functions of the eigenvalues are given and estimates of the eigenvalues are derived from these. Both a squared error loss and an entropy loss are considered. Since the resulting estimates may not be real, a method for modifying complex estimates is also given. It is shown that these modified estimates are the eigenvalues of the closest real matrix. Finally, estimation under specific eigenstructures is considered and an estimator is given that improves on any estimate of the eigenvalues in terms of risk when it is known that the smallest eigenvalue has known multiplicity. If the eigenstructure is not known, then the number of distinct eigenvalues can be estimated. If the likelihood ratio tests are used, then these tests are performed consecutively and it is shown that they are asymptotically independent. Akaike's Information Criteria (AIC) is given in this setting and shown to be inconsistent. A consistent AIC is then given. Monte Carlo studies of each of these methods are given using a variety of eigenstructures. Comparisons are made within each of the procedures and between the procedures. In particular, significant reduction in risk is possible by shrinking towards the geometric mean while less significant improvement was obtained using estimates derived from the elementary symmetric functions. Also, eigenvalue estimation after estimating the number of distinct eigenvalues offers a significant reduction in risk only for certain eigenstructures.PhDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/103607/1/9332087.pdfDescription of 9332087.pdf : Restricted to UM users only