Shrinkage estimators of covariance are an important tool in modern appliedand theoretical statistics. They play a key role in regularized estimationproblems, such as ridge regression (aka Tykhonov regularization), regularizeddiscriminant analysis and a variety of optimization problems. In this paper, we bring to bear the tools of random matrix theory tounderstand their behavior, and in particular, that of quadratic forms involvinginverses of those estimators, which are important in practice. We use very mild assumptions compared to the usual assumptions made in randommatrix theory, requiring only mild conditions on the moments of linear andquadratic forms in our random vectors. In particular, we show that our resultsapply for instance to log-normal data, which are of interest in financialapplications. Our study highlights the relative sensitivity of random matrix results (andtheir practical consequences) to geometric assumptions which are oftenimplicitly made by random matrix theorists and may not be relevant in dataanalytic practice.
展开▼
