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 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 robust
and reliable. 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
