  This paper deals with nonparametric maximum likelihood estimation for
Gaussian locally stationary processes. Our nonparametric MLE is constructed by
minimizing a frequency domain likelihood over a class of functions. The
asymptotic behavior of the resulting estimator is studied. The results depend
on the richness of the class of functions. Both sieve estimation and global
estimation are considered. Our results apply, in particular, to estimation
under shape constraints. As an example, autoregressive model fitting with a
monotonic variance function is discussed in detail, including algorithmic
considerations. A key technical tool is the time-varying empirical spectral
process indexed by functions. For this process, a Bernstein-type exponential
inequality and a central limit theorem are derived. These results for empirical
spectral processes are of independent interest.
