Simultaneous confidence bands for Yule-Walker estimators and order selection

Abstract

Let $\{X_k,k\in \mathbb{Z}\}$ be an autoregressive process of order $q$. Various estimators for the order $q$ and the parameters ${\bolds Θ}_q=(θ_1,...,θ_q)^T$ 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 ${\boldsΘ}_q$. 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 ${\max }_{1\le i\le d_n}|{\hat{\theta }}_i-{\theta }_i|$ is the Gumbel-type distribution $e^{-e^{-z}}$, where $q\in \{0,...,d_n\}$ and $d_n=\mathcal{O}(n^{\delta })$, $\delta >0$. 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 $\{{\theta }_i{\}}_{1\le i\le d_n}$ 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 $q\in \{0,...,d_n\}$ where $d_n=\mathcal{O}(n^{\delta })$.

Published in at http://dx.doi.org/10.1214/11-AOS963 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Order selection for same-realization predictions in autoregressive processes
• Revisiting two strong approximation results of Dudley and Philipp