SIMULTANEOUS CONFIDENCE BANDS FOR YULE.WALKER ESTIMATORS AND ORDER SELECTION

Let {X k , k . .} be an autoregressive process of order q. Various estimators for the order q and the parameters Q q = (.., ..., . q ) 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 Q q . In this paper, we establish simultaneous confidence bands for the Yule.Walker estimators ${\hat \theta _i}$ ; more precisely, it is shown that the limiting distribution of ${\max _{1 \leqslant i \leqslant {d_n}}}|{\hat \theta _i} - {\theta _i}|$ is the Gumbel-type distribution ${e^{ - {e^{ - z}}}}$ , where q .{0, ..., d n } and d n = O(n . ), . > 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 ${\left\{ {{\theta _i}} \right\}_{1 \leqslant i \leqslant {d_n}}}$ 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, ..., d n } where d n = O(n . ).