Public Documentation

cov(lse::LinearShrinkage, X; dims=1)

Linear shrinkage covariance estimator for matrix X along dimension dims. Computed using the method described by lse.

cov(ans::AnalyticalNonlinearShrinkage, X; dims=1, mean=nothing)

Nonlinear covariance estimator derived from the sample covariance estimator S and its eigenvalue decomposition (which can be given through decomp). See Ledoit and Wolf's paper http://www.econ.uzh.ch/static/wp/econwp264.pdf The keyword mean can be nothing (centering via estimated mean), zero (no centering) or a provided vector. In the first case, a rank-1 modification is applied and therefore the effective sample size is decreased by one (see analytical_nonlinear_shrinkage). In the latter two case the mean cannot have been estimated on the data (otherwise the effective sample size will be 1 larger than it should be resulting in numerical instabilities). If you are unsure, use either nothing or provide an explicit (non-estimated) vector (possibly a zero vector) and avoid the use of mean=0.

• Time complexity (including formation of S)
  • (p<n): O(np^2 + n^2) with moderate constant
  • (p>n): O(p^3) with low constant (dominated by eigendecomposition of S)

LinearShrinkage(target, shrinkage; corrected=false)

Linear shrinkage estimator described by equation $(1 - \lambda) S + \lambda F$ where $S$ is standard covariance matrix, $F$ is shrinkage target described by argument target and $\lambda$ is a shrinkage parameter, either given explicitly in shrinkage or automatically determined according to one of the supported methods.

The corrected estimator is used if corrected is true.

DiagonalUnitVariance

Target for linear shrinkage: unit matrix. A subtype of LinearShrinkageTarget where

• $F_{ij}=1$ if $i=j$ and
• $F_{ij}=0$ otherwise

DiagonalCommonVariance

Target for linear shrinkage: unit matrix multiplied by average variance of variables. A subtype of LinearShrinkageTarget where

• $F_{ij}=v$ if $i=j$ with $v=\mathrm{tr}(S)/p$ and
• $F_{ij}=0$ otherwise

DiagonalUnequalVariance

Target for linear shrinkage: diagonal of covariance matrix. A subtype of LinearShrinkageTarget where

• $F_{ij}=s_{ij}$ if $i=j$ and
• $F_{ij}=0$ otherwise

CommonCovariance

Target for linear shrinkage: see target_C. A subtype LinearShrinkageTarget where

• $F_{ij}=v$ if $i=j$ with $v=\mathrm{tr}(S)/p$ and
• $F_{ij}=c$ with $c=\sum_{i\neq j} S_{ij}/(p(p-1))$ otherwise

PerfectPositiveCorrelation

Target for linear shrinkage: see target_E. A subtype of LinearShrinkageTarget where

• $F_{ij}=S_{ij}$ if $i=j$ and
• $F_{ij}=\sqrt{S_{ii}S_{jj}}$ otherwise

ConstantCorrelation

Target for linear shrinkage: see target_F. A subtype of LinearShrinkageTarget where

• $F_{ij}=S_{ij}$ if $i=j$ and
• $F_{ij}=\overline{r}\sqrt{S_{ii}S_{jj}}$ otherwise where $\overline{r}$ is the average sample correlation

AnalyticalNonlinearShrinkage

Analytical nonlinear shrinkage estimator. See docs for analytical_nonlinear_shrinkage for details.