On The Probability Of Error When Using A General Akaike‐Type Criterion To Estimate Autoregression Order

. A general Akaike‐type procedure is studied where the additive penalty is proportional to the autoregression order but the constant of proportionality has a general value γ. For the procedure to be weakly consistent it is necessary and sufficient that γ→0 and nγ∝ as n∝, where n denotes the sample size. If n1‐εγ∝ for some ε>0 then the probability of erring when estimating the autoregression order converges to zero at a rate n‐c, for all c>0, as n→∝. However, several procedures suggested for practical use have n1‐εγ→0 for each ε>0; in particular, they have γ=n‐1 log n or γ=n‐1 log log n. To elucidate the properties of error probabilities in these circumstances we study the case of an AR(1) process. It is shown that in this case the probability of underestimating order is usually substantially less then the chance of overestimation, unless the autoregressive constant is particularly small (in fact, of size 2γ1/2).