The Annals of Statistics

Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes

Rainer Dahlhaus and Wolfgang Polonik

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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.

Article information

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

Dates
First available in Project Euclid: 23 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1179935065

Digital Object Identifier
doi:10.1214/009053606000000867

Mathematical Reviews number (MathSciNet)
MR2329468

Zentralblatt MATH identifier
1114.62034

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

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

Citation

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. https://projecteuclid.org/euclid.aos/1179935065


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