Simultaneous confidence bands for Yule–Walker estimators and order selection
Moritz Jirak
Source: Ann. Statist. Volume 40, Number 1
(2012), 494-528.
Abstract
Let {Xk, k ∈ ℤ} be an autoregressive process of order q. Various estimators for the order q and the parameters Θq = (θ1, …, θq)T are known; the order is usually determined with Akaike’s criterion or related modifications, whereas Yule–Walker, Burger or maximum likelihood estimators are used for the parameters Θq. In this paper, we establish simultaneous confidence bands for the Yule–Walker estimators θ̂i; more precisely, it is shown that the limiting distribution of max1≤i≤dn|θ̂i − θi| is the Gumbel-type distribution e−e−z, where q ∈ {0, …, dn} and , δ > 0. This allows to modify some of the currently used criteria (AIC, BIC, HQC, SIC), but also yields a new class of consistent estimators for the order q. These estimators seem to have some potential, since they outperform most of the previously mentioned criteria in a small simulation study. In particular, if some of the parameters {θi}1≤i≤dn are zero or close to zero, a significant improvement can be observed. As a byproduct, it is shown that BIC, HQC and SIC are consistent for q ∈ {0, …, dn} where
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Keywords: Autoregressive process; Yule–Walker estimators; extreme value distribution; order selection; AIC
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1336396181
Digital Object Identifier: doi:10.1214/11-AOS963
Zentralblatt MATH identifier: 06075623
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