Distribution-Free Shrinkage of High-Dimensional Mean Vector

We introduce shrinkage estimators of the sample mean vector in high dimension. Our estimators share desirable properties relative to existing methods: they are distribution-free, consider bona-fide target estimators, and use simple estimators of the shrinkage intensities independent of the precision matrix which are L2 consistent in high dimension. Unlike existing estimators that impose that the whitened data be an i.i.d. matrix, we only impose i.i.d. across sample observations. We require uniform boundedness of the first four moments, and the high-dimensional asymptotics are in a general Kolmogorov setting where N/T=O(1) as T→∞, with N the mean-vector dimension and T the sample size. We consider as a target estimator either zero or the grand mean. Simulations show that our shrinkage estimators are competitive with a range of benchmark estimators, both when the theoretical assumptions are satisfied and violated. Finally, we apply our estimators to constructing mean-variance portfolios of a large number of stocks, and find that they deliver robust out-of-sample Sharpe ratios.