A functional central limit theorem for regression models

Kurzfassung/Abstract

Let a linear regression model be given with an experimental region $[a,b] \subseteq\R$ and regression functions $f_1,\dots, f_{d+1}: [a,b]\to \R$. In practice it is an important question whether a certain regression function $f_{d+1}$, say, does or does not belong to the model. Therefore, we investigate the test problem $H_0$: ``$f_{d+1}$ does not belong to the model against $K$: ``$f_{d+1}$ belongs to the model based on the least-squares residuals of the observations made at design points of the experimental region $[a,b]$.\par By a new functional central limit theorem given by {\it W. Bischoff} [Ann. Stat. 26, No. 4, 1398-1410 (1998; Zbl 0936.62072)], we are able to determine optimal tests in an asymptotic way. Moreover, we introduce the problem of experimental design for the optimal test statistics. Further, we compare the asymptotically optimal test with the likelihood ratio test $(F$-test) under the assumption that the error is normally distributed. Finally, we consider real change-point problems as examples and investigate by simulations the behavior of the asymptotic test for finite sample sizes. We determine optimal designs for these examples.