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March, 1988 Testing the (Parametric) Null Model Hypothesis in (Semiparametric) Partial and Generalized Spline Models
Dennis Cox, Eunmee Koh, Grace Wahba, Brian S. Yandell
Ann. Statist. 16(1): 113-119 (March, 1988). DOI: 10.1214/aos/1176350693

Abstract

Cox and Koh (1986) considered the model $y_i = f(x(i)) + \varepsilon_i, \varepsilon_i$ i.i.d. $N(0, \sigma^2)$, with the (parametric) null hypothesis $f(x), x \in \lbrack 0, 1 \rbrack$, a polynomial of degree $m - 1$ or less, versus the alternative $f$ is "smooth," based on the Bayesian model for $f$ which leads to polynomial smoothing spline estimates for $f$. They showed that there was no uniformly most powerful test and found the locally most powerful (LMP) test. We extend their result to the generalized smoothing spline models of Wahba (1985) and to the partial spline models proposed and studied by Engle, Granger, Rice and Weiss (1986), Shiller (1984), Green, Jennison and Seheult (1985), Wahba (1984), Heckman (1986) and others. We also show that the test statistic has an intimate relationship with the behavior of the generalized cross validation (GCV) function at $\lambda = \infty$. If the GCV function has a minimum at $\lambda = \infty$, then GCV has chosen the (parametric) model corresponding to the null hypothesis; we show that if the LMP test statistic is no larger than a certain multiple of the residual sum of squares after (parametric) regression, then the GCV function will have a (possibly local) minimum at $\lambda = \infty$.

Citation

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Dennis Cox. Eunmee Koh. Grace Wahba. Brian S. Yandell. "Testing the (Parametric) Null Model Hypothesis in (Semiparametric) Partial and Generalized Spline Models." Ann. Statist. 16 (1) 113 - 119, March, 1988. https://doi.org/10.1214/aos/1176350693

Information

Published: March, 1988
First available in Project Euclid: 12 April 2007

zbMATH: 0673.62017
MathSciNet: MR924859
Digital Object Identifier: 10.1214/aos/1176350693

Subjects:
Primary: 62F03
Secondary: 62C10, 62G10, 62H15

Rights: Copyright © 1988 Institute of Mathematical Statistics

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Vol.16 • No. 1 • March, 1988
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