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Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes. (English) Zbl 1114.62034
Summary: 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.

##### MSC:
 62G05 Nonparametric estimation 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62M09 Non-Markovian processes: estimation 62F30 Parametric inference under constraints 65C60 Computational problems in statistics (MSC2010) 60F05 Central limit and other weak theorems
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