Matrix shrinkage of high-dimensional expectation vectors

The shrinkage effect is studied in estimating the expectation vector by weighting of mean vector components in the system of coordinates in which sample covariance matrix is diagonal. The Kolmogorov asymptotic approach is applied, when sample size increases together with the dimension, so that their ratio tends to a constant. Under some weak assumptions on the dependence of variables, the limit expression for the principal part of the quadratic risk function is found in dependence of weighting function. It is proved that the limit risk function does not depend on distributions. The extremum problem is solved, and an approximately unimprovable distribution-free estimator of the expectation vector is proposed. (C) 2003 Elsevier Inc. All rights reserved.