The Annals of Statistics

Maximum likelihood estimation for α-stable autoregressive processes

Beth Andrews, Matthew Calder, and Richard A. Davis

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Abstract

We consider maximum likelihood estimation for both causal and noncausal autoregressive time series processes with non-Gaussian α-stable noise. A nondegenerate limiting distribution is given for maximum likelihood estimators of the parameters of the autoregressive model equation and the parameters of the stable noise distribution. The estimators for the autoregressive parameters are n1/α-consistent and converge in distribution to the maximizer of a random function. The form of this limiting distribution is intractable, but the shape of the distribution for these estimators can be examined using the bootstrap procedure. The bootstrap is asymptotically valid under general conditions. The estimators for the parameters of the stable noise distribution have the traditional n1/2 rate of convergence and are asymptotically normal. The behavior of the estimators for finite samples is studied via simulation, and we use maximum likelihood estimation to fit a noncausal autoregressive model to the natural logarithms of volumes of Wal-Mart stock traded daily on the New York Stock Exchange.

Article information

Source
Ann. Statist., Volume 37, Number 4 (2009), 1946-1982.

Dates
First available in Project Euclid: 18 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1245332837

Digital Object Identifier
doi:10.1214/08-AOS632

Mathematical Reviews number (MathSciNet)
MR2533476

Zentralblatt MATH identifier
1168.62077

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62E20: Asymptotic distribution theory 62F10: Point estimation

Keywords
Autoregressive models maximum likelihood estimation noncausal non-Gaussian stable distributions

Citation

Andrews, Beth; Calder, Matthew; Davis, Richard A. Maximum likelihood estimation for α -stable autoregressive processes. Ann. Statist. 37 (2009), no. 4, 1946--1982. doi:10.1214/08-AOS632. https://projecteuclid.org/euclid.aos/1245332837


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