Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes

Abstract

This paper deals with nonparametric maximum likelihood estimation for\nGaussian locally stationary processes. Our nonparametric MLE is constructed by\nminimizing a frequency domain likelihood over a class of functions. The\nasymptotic behavior of the resulting estimator is studied. The results depend\non the richness of the class of functions. Both sieve estimation and global\nestimation are considered. Our results apply, in particular, to estimation\nunder shape constraints. As an example, autoregressive model fitting with a\nmonotonic variance function is discussed in detail, including algorithmic\nconsiderations. A key technical tool is the time-varying empirical spectral\nprocess indexed by functions. For this process, a Bernstein-type exponential\ninequality and a central limit theorem are derived. These results for empirical\nspectral processes are of independent interest.\n