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

Abstract

We introduce a class of Toeplitz-band matrices for simple goodness of ®t 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 ®xed 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 con®dence 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 ®t and estimating the error variance is highlighted. As a by-product we obtain a much simpler proof of a result of 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 ®nite sample approximations.

Convergence Rates of General Regularization Methods for Statistical Inverse Problems and Applications

“…We can replace L 2 (μ 1 ) by an arbitrary Hilbert space H 1 (e.g., a Sobolev space) by replacing k(x, ·) byk(x) : [36] and O'Sullivan [37]). For conditions on the design density see Munk [34].…”

An ANOVA-type nonparametric diagnostic test for heteroscedastic regression models

“…Testing for a constant regression function. In Table 3, we compare our test with a recent procedure proposed by Munk [19]. Munk's test is based on a quadratic measure of the discrepancy between the postulated parametric model and the true model, which is estimated by random Toeplitz forms.…”

“…To test for the null hypothesis of a more general parametric functional form, such as a linear regression, we may apply the same approach to residuals from the parametric fit under the null model. For more general procedures, we refer to Azzalini and Bowman [7], González Manteiga and Cao [8], Härdle and Mammen [9], Fan and Li [10], Stute [11], the monograph of Hart [12], Dette and Munk [13,14], Aït-Sahalia et al [15], Fan and Huang [16], Fan et al [17], Horowitz and Spokoiny [18] and Munk [19] and the references therein. The form of the new test is motivated by recent developments in analysis of variance with large number of factor levels [20].…”

Parametric versus non-parametric modelling? Statistical evidence based on P-value curves

“…Finally, the astrophysicist has to decide whether a value of M 2 =Π should be regarded as scientifically negligible or as a deviation from the ‘true’ density K ρ, which is considered significant for astrophysical reasons. For a more thorough introduction to P ‐value curves in an astrophysical context see BM2, and for the statistical theory see Munk (2002), and references given therein.…”