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A difference-based method for testing no effect in nonparametric regression

Author

Listed:
  • Zhijian Li

    (BNU-HKBU United International College)

  • Tiejun Tong

    (Hong Kong Baptist University)

  • Yuedong Wang

    (University of California)

Abstract

The paper proposes a novel difference-based method for testing the hypothesis of no relationship between the dependent and independent variables. We construct three test statistics for nonparametric regression with Gaussian and non-Gaussian random errors. These test statistics have the standard normal as the asymptotic null distribution. Furthermore, we show that these tests can detect local alternatives that converge to the null hypothesis at a rate close to $$n^{-1/2}$$ n - 1 / 2 previously achieved only by the residual-based tests. We also propose a permutation test as a flexible alternative. Our difference-based method does not require estimating the mean function or its first derivative, making it easy to implement and computationally efficient. Simulation results demonstrate that our new tests are more powerful than existing methods, especially when the sample size is small. The usefulness of the proposed tests is also illustrated using two real data examples.

Suggested Citation

  • Zhijian Li & Tiejun Tong & Yuedong Wang, 2025. "A difference-based method for testing no effect in nonparametric regression," Computational Statistics, Springer, vol. 40(1), pages 153-176, January.
  • Handle: RePEc:spr:compst:v:40:y:2025:i:1:d:10.1007_s00180-024-01479-0
    DOI: 10.1007/s00180-024-01479-0
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    References listed on IDEAS

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    1. Wenceslao González-Manteiga & Rosa Crujeiras, 2013. "Rejoinder on: 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 442-447, September.
    2. Dennis Cox & Eunmee Koh, 1989. "A smoothing spline based test of model adequacy in polynomial regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 41(2), pages 383-400, June.
    3. 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.
    4. McManus, Douglas A., 1991. "Who Invented Local Power Analysis?," Econometric Theory, Cambridge University Press, vol. 7(2), pages 265-268, June.
    5. R. L. Eubank, 2000. "Testing for No Effect by Cosine Series Methods," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 747-763, December.
    6. Yatchew, A., 1999. "An elementary nonparametric differencing test of equality of regression functions," Economics Letters, Elsevier, vol. 62(3), pages 271-278, March.
    7. Tiejun Tong & Yuedong Wang, 2005. "Estimating residual variance in nonparametric regression using least squares," Biometrika, Biometrika Trust, vol. 92(4), pages 821-830, December.
    8. Einmahl, J.H.J. & van Keilegom, I., 2006. "Tests for Independence in Nonparametric Regression," Other publications TiSEM 0c6f2c43-aa7d-45c1-9d43-7, Tilburg University, School of Economics and Management.
    9. Yatchew,Adonis, 2003. "Semiparametric Regression for the Applied Econometrician," Cambridge Books, Cambridge University Press, number 9780521812832, June.
    10. Ingrid Keilegom & Wenceslao González Manteiga & César Sánchez Sellero, 2008. "Goodness-of-fit tests in parametric regression based on the estimation of the error distribution," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(2), pages 401-415, August.
    11. Juei-Chao Chen, 1994. "Testing for no effect in nonparametric regression via spline smoothing techniques," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(2), pages 251-265, June.
    12. Chin-Shang Li, 2012. "Testing for no effect via splines," Computational Statistics, Springer, vol. 27(2), pages 343-357, June.
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