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Testing the Goodness of Fit of Parametric Regression Models with Random Toeplitz Forms[Note 1. Dedicated to Frits Ruymgaart on the occasion of his ...]
Author(s) -
MUNK AXEL
Publication year - 2002
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/1467-9469.00303
Subject(s) - mathematics , goodness of fit , parametric statistics , test statistic , statistics , toeplitz matrix , estimator , asymptotic distribution , parametric model , statistical hypothesis testing , pure mathematics
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.