# 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 |
---|---|---|---|

`DiagonalUnitVariance` | $1$ | $0$ | $F = \mathbf I$ |

`DiagonalCommonVariance` | $v$ | 0 | $F = v\mathbf I$ |

`DiagonalUnequalVariance` | $S_{ii}$ | 0 | $F = \mathrm{diag}(S)$, very common |

`CommonCovariance` | $v$ | $c$ | |

`PerfectPositiveCorrelation` | $S_{ii}$ | $\sqrt{S_{ii}S_{jj}}$ | |

`ConstantCorrelation` | $S_{ii}$ | $\overline{r}\sqrt{S_{ii}S_{jj}}$ | 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.