Simultaneous Confidence Bands for Yule–walker Estimators and Order Selection by Moritz Jirak

Let {Xk, k ∈ Z} be an autoregressive process of order q. Various estimators for the order q and the parameters q = (θ1, . . . , θq) 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 q . 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 , where q ∈ {0, . . . , dn} and dn =O(nδ), 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 dn =O(nδ).