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

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