The Annals of Statistics

Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes

Rainer Dahlhaus and Wolfgang Polonik

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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.

Article information

Ann. Statist., Volume 34, Number 6 (2006), 2790-2824.

First available in Project Euclid: 23 May 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62F30: Inference under constraints

Empirical spectral process exponential inequalities for quadratic forms nonparametric maximum likelihood estimation locally stationary processes sieve estimation


Dahlhaus, Rainer; Polonik, Wolfgang. Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes. Ann. Statist. 34 (2006), no. 6, 2790--2824. doi:10.1214/009053606000000867.

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