Simultaneous confidence bands for sequential autoregressive fitting

Let { X k , 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 Θ ̂ m : = ( θ ̂ m , 1 , … , θ ̂ m , m ) ⊤ , m = 1 , 2 , … .. Under mild conditions, simultaneous confidence bands for Θ ̂ m , Θ ̂ m + 1 , … are derived. More precisely, it is shown that max d n − κ n ≤ m ≤ d n max 1 ≤ h ≤ m | θ ̂ m , h − θ h | converges to an extreme value distribution, where d n = O ( n δ ) , δ > 0 , and n denotes the sample size. The relation of κ n and d n depends on the bias term ∑ i = d n − 2 κ n ∞ | θ i | . 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.