Testing the goodness of fit of parametric regression models with random Toeplitz forms

We introduce a class of Toeplitz-band matrices for simple goodness of fit tests for parametric regression models. For a given length r of the band matrix the asymptotic optimal solution is derived. Asymptotic normality of the corresponding test statistic is established under a fixed and random design assumption as well as for linear and non-linear models, respectively. This allows testing at any parametric assumption as well as the computation of confidence intervals for a quadratic measure of discrepancy between the parametric model and the true signal g. Furthermore, the connection between testing the parametric goodness of fit and estimating the error variance is highlighted. As a by-product we obtain a much simpler proof of a result of Hall et al. (1990) concerning the optimality of an estimator for the variance. Our results unify and generalize recent results by Brodeau (1993) and Dette & Munk (1998a,b) in several directions. Extensions to multivariate predictors and unbounded signals are discussed. A simulation study shows that a simple jacknife correction of the proposed test statistics leads to reasonable finite sample approximations.