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August 2009 Maximum likelihood estimation for α-stable autoregressive processes
Beth Andrews, Matthew Calder, Richard A. Davis
Ann. Statist. 37(4): 1946-1982 (August 2009). DOI: 10.1214/08-AOS632


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.


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Beth Andrews. Matthew Calder. Richard A. Davis. "Maximum likelihood estimation for α-stable autoregressive processes." Ann. Statist. 37 (4) 1946 - 1982, August 2009.


Published: August 2009
First available in Project Euclid: 18 June 2009

zbMATH: 1168.62077
MathSciNet: MR2533476
Digital Object Identifier: 10.1214/08-AOS632

Primary: 62M10
Secondary: 62E20 , 62F10

Keywords: autoregressive models , maximum likelihood estimation , noncausal , non-Gaussian , Stable distributions

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 4 • August 2009
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