  Much research has been carried out on shrinkage methods for real-valued
covariance matrices. In spectral analysis of $p$-vector-valued time series
there is often a need for good shrinkage methods too, most notably when the
complex-valued spectral matrix is singular. The equivalent of the Ledoit-Wolf
(LW) covariance matrix estimator for spectral matrices can be improved on using
a Rao-Blackwell estimator, and using random matrix theory we derive its form.
Such estimators can be used to better estimate inverse spectral (precision)
matrices too, and a random matrix method has previously been proposed and
implemented via extensive simulations. We describe the method, but carry out
computations entirely analytically, and suggest a way of selecting an important
parameter using a predictive risk approach. We show that both the Rao-Blackwell
estimator and the random matrix estimator of the precision matrix can
substantially outperform the inverse of the LW estimator in a time series
setting. Our new methodology is applied to EEG-derived time series data where
it is seen to work well and deliver substantial improvements for precision
matrix estimation.
