CovTools (version 0.5.4)

CovEst.2010RBLW: Rao-Blackwell Ledoit-Wolf Estimator

Description

Authors propose to estimate covariance matrix by minimizing mean squared error with the following formula, $$\hat{\Sigma} = \rho \hat{F} + (1-\rho) \hat{S}$$ where \(\rho \in (0,1)\) a control parameter/weight, \(\hat{S}\) an empirical covariance matrix, and \(\hat{F}\) a target matrix. It is proposed to use a structured estimate \(\hat{F} = \textrm{Tr} (\hat{S}/p) \cdot I_{p\times p}\) where \(I_{p\times p}\) is an identity matrix of dimension \(p\).

Usage

CovEst.2010RBLW(X)

Arguments

X

an \((n\times p)\) matrix where each row is an observation.

Value

a named list containing:

S

a \((p\times p)\) covariance matrix estimate.

rho

an estimate for convex combination weight.

References

chen_shrinkage_2010CovTools

Examples

# NOT RUN {
## CRAN-purpose small computation
# set a seed for reproducibility
set.seed(11)

#  small data with identity covariance
pdim      <- 10
dat.small <- matrix(rnorm(5*pdim), ncol=pdim)

#  run the code
out.small <- CovEst.2010RBLW(dat.small)

#  visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
image(diag(pdim)[,pdim:1],     main="true cov")
image(cov(dat.small)[,pdim:1], main="sample cov")
image(out.small$S[,pdim:1],    main="estimated cov")
par(opar)

# }
# NOT RUN {
## want to see how delta is determined according to
#  the number of observations we have.
nsamples = seq(from=5, to=200, by=5)
nnsample = length(nsamples)

#  we will record two values; rho and norm difference
vec.rho   = rep(0, nnsample)
vec.normd = rep(0, nnsample)
for (i in 1:nnsample){
  dat.norun <- matrix(rnorm(nsamples[i]*pdim), ncol=pdim) # sample in R^5
  out.norun <- CovEst.2010RBLW(dat.norun)                 # run with default

  vec.rho[i]   = out.norun$rho
  vec.normd[i] = norm(out.norun$S - diag(5),"f")          # Frobenius norm
}

# let's visualize the results
opar <- par(mfrow=c(1,2))
plot(nsamples, vec.rho,   lwd=2, type="b", col="red", main="estimated rhos")
plot(nsamples, vec.normd, lwd=2, type="b", col="blue",main="Frobenius error")
par(opar)
# }
# NOT RUN {
# }