| Abstract: | ABSTRACT: 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 34 ) concerning the optimality of an estimator for the variance. Our results unify and generalize recent results by 9 ) and 15 , 16 ) 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. |
