On Designs for Recursive Least Squares Residuals to Detect Alternatives

TL;DR

This paper investigates the asymptotic behavior of recursive least squares residuals in sequential regression testing, establishing limit processes under the null hypothesis and local alternatives, and identifying optimal alternative designs for enhanced test power.

Contribution

It derives the limit processes for recursive residuals in sequential regression and determines optimal alternative designs to improve test power.

Findings

Limit process of recursive residuals under null hypothesis

Limit process under local alternatives

Identification of optimal alternative designs for test power

Abstract

Linear regression models are checked by a lack-of-fit (LOF) test to be sure that the model is at least approximatively true. In many practical cases data are sampled sequentially. Such a situation appears in industrial production when goods are produced one after the other. So it is of some interest to check the regression model sequentially. This can be done by recursive least squares residuals. A sequential LOF test can be based on the recursive residual partial sum process. In this paper we state the limit of the partial sum process of a triangular array of recursive residuals given a constant regression model when the number of observations goes to infinity. Furthermore, we state the corresponding limit process for local alternatives. For specific alternatives designs are determined dominating other designs in respect of power of the sequential LOF test described above. In this context a result is given in which $e^{-1}$ plays a crucial role.