Theory & Methods: ε‐Repetitions of the Maximum Residuals in an AR(1) Model

Abstract

This paper considers record values of residuals or prediction errors in a one‐parameter autoregressive process and the statistic Zn= number of ε‐repetitions of this record. When the parameter of the autoregression is unknown, the prediction errors, and therefore Zn, are unobservable. Here an observable analogue Ẑ of Zn is considered. It is proved that under special conditions the difference Zn− unobservable. Here an observable analogue Ẑ converges to zero in probability and therefore that unobservable. Here an observable analogue Ẑ has the same asymptotic behaviour as Zn.