Simultaneous confidence bands for sequential autoregressive fitting

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,…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)dn=O(nδ), δ>0δ>0, and nn denotes the sample size. The relation of κnκn and dndn 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.