Simultaneous confidence bands for Yule–Walker estimators and order selection

Abstract

Let {X_k, k ∈ ℤ} be an autoregressive process of order q. Various estimators for the order q and the parameters Θ_q = (θ₁, …, 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 Θ_q. In this paper, we establish simultaneous confidence bands for the Yule–Walker estimators θ̂_i; more precisely, it is shown that the limiting distribution of max_1≤i≤dn|θ̂_i − θ_i| is the Gumbel-type distribution e^−e−z, where q ∈ {0, …, d_n} and $d_{n}=\mathcal{O}(n^{\delta})$, 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 {θ_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 ∈ {0, …, d_n} where $d_{n}=\mathcal{O}(n^{\delta})$.