  A hybrid estimator of the log-spectral density of a stationary time series is
proposed. First, a multiple taper estimate is performed, followed by kernel
smoothing the log-multitaper estimate. This procedure reduces the expected mean
square error by $({\pi^2 \over 4})^{.8}$ over simply smoothing the log tapered
periodogram. The optimal number of tapers is $O(N^{8/15})$. A data adaptive
implementation of a variable bandwidth kernel smoother is given. When the
spectral density is discontinuous, one sided smoothing estimates are used.
