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December 2006 Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes
Rainer Dahlhaus, Wolfgang Polonik
Ann. Statist. 34(6): 2790-2824 (December 2006). DOI: 10.1214/009053606000000867


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.


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Rainer Dahlhaus. Wolfgang Polonik. "Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes." Ann. Statist. 34 (6) 2790 - 2824, December 2006.


Published: December 2006
First available in Project Euclid: 23 May 2007

zbMATH: 1114.62034
MathSciNet: MR2329468
Digital Object Identifier: 10.1214/009053606000000867

Primary: 62M10
Secondary: 62F30

Keywords: empirical spectral process , exponential inequalities for quadratic forms , Locally stationary processes , nonparametric maximum likelihood estimation , sieve estimation

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 6 • December 2006
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