Testing the Goodness of Fit of Parametric Regression Models with Random Toeplitz Forms
Axel Munk · Scandinavian Journal of Statistics · 2002
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 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.