Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes

TL;DR

This paper introduces a nonparametric maximum likelihood estimation method for Gaussian locally stationary processes, analyzing its asymptotic properties and applying it to shape-constrained autoregressive model fitting.

Contribution

It develops a novel nonparametric MLE framework using frequency domain likelihood, with theoretical analysis and practical algorithms for shape-constrained estimation.

Findings

Asymptotic normality of the estimator established

Effective algorithms for shape-constrained estimation provided

Empirical spectral process inequalities derived

Abstract

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.