Order Selection, Stochastic Complexity and Kullback-Leibler Information

The main motivation for Hannan and Quinn’s ϕ criterion is the minimality of its log log n penalty rate, which is the slowest one compatible with consistency. Unlike Akaike’s AIC, based on the minimization of some estimated mean Kullback-Leibler distance, or Rissanen and Schwarz’s BIC,which minimizes an estimate of the expected complexity, Hannan and Quinn’s ϕ at first sight is not connected with any sound decision-theoretic or statistical principle. The objective of this paper is to provide a stochastic complexity justification and an information-theoretic derivation for ϕ Two generalized AIC — BIC — ϕ criteria also are proposed. Both are derived from Kullback-Leibler distance arguments, and enjoy the same consistency properties as BIC (strong consistency) and ϕ (weak consistency), respectively.