SIMULTANEOUS CONFIDENCE BANDS FOR YULE-WALKER ESTIMATORS AND ORDER SELECTION

• 成果类型: Article
• 署名作者: Jirak, Moritz
• 署名单位: Graz University of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS963
• 发表日期: 2012
• 页码: 494-528
• 关键词: Asymptotic Theory; MODEL; matrices

摘要

Let {X-k, k is an element of Z} be an autoregressive process of order q. Various estimators for the order q and the parameters Theta(q) = (theta(1), ... , theta(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 Theta(q). In this paper, we establish simultaneous confidence bands for the Yule-Walker estimators (theta) over cap (i); more precisely, it is shown that the limiting distribution of maxi (1 <= i <= dn)vertical bar(theta) over cap (i) - theta(i)vertical bar is the Gumbel-type distribution e(-e-z) where q is an element of {0, ... , d(n)} and d(n) = 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 <= 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 is an element of (0, ... , d(n)} where d(n) = O(n(delta)).