# Linear shrinkage estimators

Linear shrinkage estimators correspond to covariance estimators of the form

$$$\hat\Sigma = (1-\lambda)S + \lambda F$$$

where $F$ is a target matrix of appropriate dimensions, $\lambda\in[0,1]$ is a shrinkage intensity and $S$ is the sample covariance estimator (corrected or uncorrected depending on the corrected keyword).

## Targets and intensities

There are several standard target matrices (denoted by $F$) that can be used (we follow here the notations and naming conventions of Schaffer & Strimmer 2005):

Target name$F_{ii}$$F_{ij} (i\neq j)Comment DiagonalUnitVariance1$$0$$F = \mathbf I DiagonalCommonVariancev0F = v\mathbf I DiagonalUnequalVarianceS_{ii}0F = \mathrm{diag}(S), very common CommonCovariancev$$c$
PerfectPositiveCorrelation$S_{ii}$$\sqrt{S_{ii}S_{jj}} ConstantCorrelationS_{ii}$$\overline{r}\sqrt{S_{ii}S_{jj}}$used in Ledoit & Wolf 2004

where $v = \mathrm{tr}(S)/p$ is the average variance, $c = \sum_{i\neq j} S_{ij}/(p(p-1))$ is the average of off-diagonal terms of $S$ and $\overline{r}$ is the average of sample correlations (see Schaffer & Strimmer 2005).

For each of these targets, an optimal shrinkage intensity $\lambda^\star$ can be computed. A standard approach is to apply the Ledoit-Wolf formula (shrinkage=:lw, see Ledoit & Wolf 2004) though there are some variants that can be applied too. Notably, Schaffer & Strimmer's variant (shrinkage=:ss) will ensure that the $\lambda^\star$ computed is the same for $X_c$ (the centered data matrix) as for $X_s$ (the standardised data matrix). See Schaffer & Strimmer 2005.

Chen's variant includes a Rao-Blackwellised estimator (shrinkage=:rblw) and an Oracle-Approximating one (shrinkage=:oas) for the DiagonalCommonVariance target. See Chen, Wiesel, Eldar & Hero 2010.