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Optimal designs for testing the functional form of a regression via nonparametric estimation techniques

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  • Biedermann, Stefanie
  • Dette, Holger

Abstract

For the problem of checking linearity in a heteroscedastic nonparametric regression model under a fixed design assumption, we study maximin designs which maximize the minimum power of a nonparametric test over a broad class of alternatives from the assumed linear regression model. It is demonstrated that the optimal design depends sensitively on the used estimation technique (i.e. weighted or ordinary least-squares) and on an inner product used in the definition of the class of alternatives. Our results extend and put recent findings of Wiens (Statist. Probab. Lett. 12 (1991) 217) in a new light, who established the maximin optimality of the uniform design for lack-of-fit tests in homoscedastic multiple linear regression models.

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  • Biedermann, Stefanie & Dette, Holger, 2001. "Optimal designs for testing the functional form of a regression via nonparametric estimation techniques," Statistics & Probability Letters, Elsevier, vol. 52(2), pages 215-224, April.
    • Handle: RePEc:eee:stapro:v:52:y:2001:i:2:p:215-224
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    References listed on IDEAS

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    1. Dette, Holger, 2000. "On a nonparametric test for linear relationships," Statistics & Probability Letters, Elsevier, vol. 46(3), pages 307-316, February.
    2. W. González-Manteiga & R. Cao, 1993. "Testing the hypothesis of a general linear model using nonparametric regression estimation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 2(1), pages 161-188, December.
    3. Wiens, Douglas P., 1991. "Designs for approximately linear regression: two optimality properties of uniform designs," Statistics & Probability Letters, Elsevier, vol. 12(3), pages 217-221, September.
    4. Alcalá, J. T. & Cristóbal, J. A. & González-Manteiga, W., 1999. "Goodness-of-fit test for linear models based on local polynomials," Statistics & Probability Letters, Elsevier, vol. 42(1), pages 39-46, March.
    5. John Xu Zheng, 1996. "A consistent test of functional form via nonparametric estimation techniques," Journal of Econometrics, Elsevier, vol. 75(2), pages 263-289, December.
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    Citations

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    1. Bischoff, Wolfgang & Miller, Frank, 2006. "Lack-of-fit-efficiently optimal designs to estimate the highest coefficient of a polynomial with large degree," Statistics & Probability Letters, Elsevier, vol. 76(15), pages 1701-1704, September.
    2. Dette, Holger & Wiens, Douglas P., 2007. "Robust designs for 3D shape analysis with spherical harmonic descriptors," Technical Reports 2007,12, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    3. Linglong Kong & Douglas P. Wiens, 2015. "Model-Robust Designs for Quantile Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 233-245, March.
    4. Wiens, Douglas P., 2010. "Robustness of design for the testing of lack of fit and for estimation in binary response models," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3371-3378, December.
    5. Douglas P. Wiens, 2009. "Robust discrimination designs," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(4), pages 805-829, September.
    6. Renata Eirini Tsirpitzi & Frank Miller & Carl-Fredrik Burman, 2023. "Robust optimal designs using a model misspecification term," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(7), pages 781-804, October.
    7. Mandal, Nripes Kumar & Pal, Manisha, 2013. "Maximin designs for the detection of synergistic effects," Statistics & Probability Letters, Elsevier, vol. 83(7), pages 1632-1637.

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