Ledoit-Wolf linear shrinkage with unknown mean

Abstract: This work addresses large dimensional covariance matrix estimation with unknown mean. The empirical covariance estimator fails when dimension and number of samples are proportional and tend to infinity, settings known as Kolmogorov asymptotics. When the mean is known, Ledoit and Wolf (2004) proposed a linear shrinkage estimator and proved its convergence under those asymptotics. To the best of our knowledge, no formal proof has been proposed when the mean is unknown. To address this issue, we propose to extend the linear shrinkage and its convergence properties to translation-invariant estimators. We expose four estimators respecting those conditions, proving their properties. Finally, we show empirically that a new estimator we propose outperforms other standard estimators.