Asymptotic Equivalence between Information- and Prediction-Based Model Selection Criteria: A New Look at AIC

• Activity Number: 189
• Type: Invited
• Date/Time: Tuesday, August 5, 2003 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Ching-Kang Ing*+
• Companies: Institute of Statistical Science, Academia Sinica
• Keywords: AIC ; accumulated prediction errors ; asymptotic efficiency ; asymptotic equivalence ; autoregressive process ; model selection

Abstract

We investigate the predictive ability of the accumulated prediction error (APE) of Rissanen in an infinite-order autoregressive (AR) model. While APE is asymptotically equivalent to BIC and is not asymptotically efficient in an infinite-order AR model, a modification of APE, APE_{w}, is proposed to rectify this difficulty. Instead of accumulating sequential prediction errors from the beginning, APE_{w} is obtained by accumulating sequential prediction errors from stage nw, where n is the sample size and 0 < w < 1. Under certain regularity conditions, we show that APE_{w} is asymptotically efficient in the sense that the mean-squared prediction error of the AR model with order selected by APE_{w} can ultimately achieve the best compromise between model complexity and the goodness of fit. Based on this result, we further show that APE_{w} is asymptotically equivalent to AIC in an infinite-order AR model. This is a somewhat interesting discovery because the proposed modification changes the nature of APE from a BIC-like criterion to an AIC-like criterion, which also provides a new look at AIC.