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Testing the Goodness of Fit of Parametric Regression Models with Random Toeplitz Forms

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  • AXEL MUNK

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 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.

Suggested Citation

  • Axel Munk, 2002. "Testing the Goodness of Fit of Parametric Regression Models with Random Toeplitz Forms," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 29(3), pages 501-533, September.
    • Handle: RePEc:bla:scjsta:v:29:y:2002:i:3:p:501-533
    DOI: 10.1111/1467-9469.00303
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    1. Bissantz, Nicolai & Hohage, T. & Munk, Axel & Ruymgaart, F., 2007. "Convergence rates of general regularization methods for statistical inverse problems and applications," Technical Reports 2007,04, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    2. Dette, Holger & Hetzler, Benjamin, 2006. "A simple test for the parametric form of the variance function in nonparametric regression," Technical Reports 2006,07, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    3. Ellingson, Leif & Patrangenaru, Vic & Ruymgaart, Frits, 2013. "Nonparametric estimation of means on Hilbert manifolds and extrinsic analysis of mean shapes of contours," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 317-333.
    4. Holger Dette & Benjamin Hetzler, 2009. "A simple test for the parametric form of the variance function in nonparametric regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(4), pages 861-886, December.

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