Partial coherence is an important quantity derived
from spectral or precision matrices and is used in seismology,
meteorology, oceanography, neuroscience and elsewhere. If the
number of complex degrees of freedom only slightly exceeds
the dimension of the multivariate stationary time series, spectral
matrices are poorly conditioned and shrinkage techniques suggest
themselves.When true partial coherencies are quite large then for
shrinkage estimators of the diagonal weighting kind it is shown
empirically that the minimization of risk using quadratic loss
(QL) leads to oracle partial coherence estimators far superior
to those derived by minimizing risk using Hilbert-Schmidt (HS)
loss. When true partial coherencies are small the methods behave
similarly. We derive two new QL estimators for spectral matrices,
and new QL and HS estimators for precision matrices. In addition
for the full estimation (non-oracle) case where certain trace
expressions must also be estimated, we examine the behaviour of
three different QL estimators, the precision matrix one seeming
particularly appealing. For the empirical study we carry out
exact simulations derived from real EEG data for two individuals,
one having large, and the other small, partial coherencies. This
ensures our study covers cases of real-world relevance.