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Simultaneous confidence bands for sequential autoregressive fitting

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  • Jirak, Moritz

Abstract

Let {Xk,k∈Z} be a zero mean causal AR(∞) process with parameter Θ∈R∞. A very common fitting procedure is to employ the Yule–Walker equations in connection with the Durbin–Levinson algorithm, which yields the (recursive) sequence of estimators Θ̂m:=(θ̂m,1,…,θ̂m,m)⊤, m=1,2,….. Under mild conditions, simultaneous confidence bands for Θ̂m, Θ̂m+1,… are derived. More precisely, it is shown that maxdn−κn≤m≤dnmax1≤h≤m|θ̂m,h−θh| converges to an extreme value distribution, where dn=O(nδ), δ>0, and n denotes the sample size. The relation of κn and dn depends on the bias term ∑i=dn−2κn∞|θi|. This significantly extends a recent result in Jirak (2012). Moreover, extensions of results of An et al. (1982) and Bhansali (1978) are obtained. In addition, the behavior of Information criteria in the AR(∞) setting is briefly discussed.

Suggested Citation

  • Jirak, Moritz, 2014. "Simultaneous confidence bands for sequential autoregressive fitting," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 130-149.
    • Handle: RePEc:eee:jmvana:v:124:y:2014:i:c:p:130-149
    DOI: 10.1016/j.jmva.2013.10.018
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