On detecting alternatives by one-parametric recursive residuals

We consider a linear regression model with one unknown parameter which is estimated by the least squares method. We suppose that, in reality, the given observations satisfy a close alternative to the linear regression model. We investigate the limiting behaviour of the normalized process of sums of recursive residuals. Such residuals were introduced by Brown, Durbin and Evans (1975) and their sums are a convenient tool for detecting discrepancy between observations and the studied model. In particular, under less restrictive assumptions we generalize a key result from Bischoff (2016).