  We develop unified theory and methodology for the inference of evolutionary
Fourier power spectra for a general class of locally stationary and possibly
nonlinear processes. In particular, simultaneous confidence regions (SCR) with
asymptotically correct coverage rates are constructed for the evolutionary
spectral densities on a nearly optimally dense grid of the joint time-frequency
domain. A simulation based bootstrap method is proposed to implement the SCR.
The SCR enables researchers and practitioners to visually evaluate the
magnitude and pattern of the evolutionary power spectra with asymptotically
accurate statistical guarantee. The SCR also serves as a unified tool for a
wide range of statistical inference problems in time-frequency analysis ranging
from tests for white noise, stationarity and time-frequency separability to the
validation for non-stationary linear models.
