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February 2012 Simultaneous confidence bands for Yule–Walker estimators and order selection
Moritz Jirak
Ann. Statist. 40(1): 494-528 (February 2012). DOI: 10.1214/11-AOS963


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≤idn|θ̂iθi| is the Gumbel-type distribution eez, where q ∈ {0, …, dn} and $d_{n}=\mathcal{O}(n^{\delta})$, 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≤idn 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 $d_{n}=\mathcal{O}(n^{\delta})$.


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Moritz Jirak. "Simultaneous confidence bands for Yule–Walker estimators and order selection." Ann. Statist. 40 (1) 494 - 528, February 2012.


Published: February 2012
First available in Project Euclid: 7 May 2012

zbMATH: 1246.62187
MathSciNet: MR3014315
Digital Object Identifier: 10.1214/11-AOS963

Primary: 60M10 , 62F05
Secondary: 62F10 , 62F12

Keywords: AIC , autoregressive process , extreme value distribution , order selection , Yule–Walker estimators

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 1 • February 2012
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