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A nonparametric least-squares test for checking a polynomial relationship

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  • Gijbels, Irène
  • Rousson, Valentin

Abstract

In this paper the interest is in testing whether a regression function is a polynomial of a certain degree. One possible approach to this testing problem is to do a parametric polynomial fit and a nonparametric fit and to reject the null hypothesis of a polynomial function if the distance between the two fits is too large. Another approach consists of looking at the residuals from the parametric fit. In this paper we propose an entirely new approach to deal with the testing problem. When testing whether a regression function is a polynomial of degree smaller than or equal to p, the key idea is to first obtain a nonparametric local polynomial estimate of the pth derivative of the unknown regression function, and then to proceed with a classical least-squares test for a general linear model for testing whether this derivative is constant. This is a quite appealing approach since it just relies on ordinary least-squares tests, and hence is simple to use. The performance of the method is illustrated via a simulation study.

Suggested Citation

  • Gijbels, Irène & Rousson, Valentin, 2001. "A nonparametric least-squares test for checking a polynomial relationship," Statistics & Probability Letters, Elsevier, vol. 51(3), pages 253-261, February.
    • Handle: RePEc:eee:stapro:v:51:y:2001:i:3:p:253-261
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Mario Francisco-Fernández & Juan Vilar-Fernández, 2009. "Two tests for heteroscedasticity in nonparametric regression," Computational Statistics, Springer, vol. 24(1), pages 145-163, February.
    2. Wenceslao González-Manteiga & Rosa Crujeiras, 2013. "An updated review of Goodness-of-Fit tests for regression models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 361-411, September.
    3. J. Opsomer & M. Francisco-Fernández, 2010. "Finding local departures from a parametric model using nonparametric regression," Statistical Papers, Springer, vol. 51(1), pages 69-84, January.
    4. Elshkaki, Ayman & van der Voet, Ester & Timmermans, Veerle & Van Holderbeke, Mirja, 2005. "Dynamic stock modelling: A method for the identification and estimation of future waste streams and emissions based on past production and product stock characteristics," Energy, Elsevier, vol. 30(8), pages 1353-1363.

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