Strong consistency of the PLS criterion for order determination of autoregressive processes. (English) Zbl 0675.62061

Hemerly, E. M.; Davis, M. H. A.

Ann. Stat. 17, No. 2, 941-946 (1989).

Summary: This note concerns the problem of order determination for autoregressive models. J. Rissanen’s, “predictive least-squares principle” [IMA J. Math. Control Inf. 3, 211-222 (1986; Zbl 0626.93069)] prescribes that one should choose as order estimate \(\hat k(n)\) at time n the order of the model which has given the least mean square prediction error up to that time. We show that this procedure is strongly consistent, that is, that \(\hat k(n)\to p\) a.s. as \(n\to \infty\) when the data are generated by an AR process of order p, given an upper bound \(p^*\).

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
93E12	Identification in stochastic control theory

Keywords:

martingale difference; strong consistency; structure identification; order determination; autoregressive models; predictive least-squares principle; least mean square prediction error

Citations:

Zbl 0626.93069