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Test for Heteroscedasticity in Partially Linear Regression Models

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Abstract

Testing heteroscedasticity determines whether the regression model can predict the dependent variable consistently across all values of the explanatory variables. Since the proposed tests could not detect heteroscedasticity in all cases, more precisely in heavy-tailed distributions, the authors established new comprehensive test statistic based on Levene’s test. The authors built the asymptotic normality of the test statistic under the null hypothesis of homoscedasticity based on the recent theory of analysis of variance for the infinite factors level. The proposed test uses the residuals from a regression model fit of the mean function with Levene’s test to assess homogeneity of variance. Simulation studies show that our test yields better than other methods in almost all cases even if the variance is a nonlinear function. Finally, the proposed method is implemented through a real data-set.

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References

  1. Engle R F, Granger C W, Rice J, et al., Semiparametric estimates of the relation between weather and electricity sales, Journal of the American statistical Association, 1986, 81(394): 310–320.

    Article  Google Scholar 

  2. Härdle W, Liang H, and Gao J T, Partially Linear Models, Springer-Physica-Verlag, Heidelberg, 2000.

    Book  MATH  Google Scholar 

  3. Greene W H, Econometric Analysis, 7th Edition, Prentice Hall, New Jersey, 2011.

    Google Scholar 

  4. Ruppert D, Wand M P, and Carroll R J, Semiparametric Regression, Cambridge University Press, Cambridge, 2003.

    Book  MATH  Google Scholar 

  5. Stone J C, Additive regression and other nonparametric models, Annals of Statistics, 1985, 13: 689–705.

    Article  MathSciNet  MATH  Google Scholar 

  6. Hastie T J and Tibshirani R J, Generalized Additive Models, Monographs on Statistics and Applied Probability, Chapman and Hall, London, 1990.

    Google Scholar 

  7. Härdle W, Mori Y, and Vieu P, Statistical Methods for Biostatistics and Related Fields, Berlin, 2006.

  8. Dette H and Munk A, Testing heteroscedasticity in nonparametric regression, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 1998, 60(4): 693–708.

    Article  MathSciNet  MATH  Google Scholar 

  9. Dette H, A consistent test for heteroscedasticity in nonparametric regression based on the kernel method, Journal of Statistical Planning and Inference, 2002, 103(1-2): 311–329.

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang L and Zhou X H, A fully nonparametric diagnostic test for homogeneity of variances, Canadian Journal of Statistics, 2005, 33: 545–558.

    Article  MathSciNet  MATH  Google Scholar 

  11. Dette H and Hetzler B, A simple test for the parametric form of the variance function in non-parametric regression, Annals of the Institute of Statistical Mathematics, 2009, 61(4): 861–886.

    Article  MathSciNet  MATH  Google Scholar 

  12. Diblasi A and Bowman A, Testing for constant variance in a linear model, Statistics & Probability Letters, 1997, 33(1): 95–103.

    Article  MATH  Google Scholar 

  13. Dette H, Neumeyer N, and Keilegom I V, A new test for the parametric form of the variance function in non-parametric regression, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2007, 69(5): 903–17.

    Article  MathSciNet  Google Scholar 

  14. Francisco-Fernndez M and Vilar-Fernndez J M, Two tests for heteroscedasticity in nonparametric regression, Computational Statistics, 2009, 24(1): 145–163.

    Article  MathSciNet  Google Scholar 

  15. Zheng X, Testing heteroscedasticity in nonlinear and nonparametric regressions, The Canadian Journal of Statistics, 2009, 37: 282–300.

    Article  MathSciNet  MATH  Google Scholar 

  16. Dette H and Marchlewski M, A robust test for homoscedasticity in nonparametric regression, Journal of Nonparametric Statistics, 2010, 22(6): 723–36.

    Article  MathSciNet  MATH  Google Scholar 

  17. Levene H, Robust tests for equality of variances, Probability and Statistics, Essays in Honor of Harold Hotelling, Stanford University Press, Stanford, 1960, 278–292.

    Google Scholar 

  18. Akritas M G and Papadatos N, Heteroscedastic one-way ANOVA and lack-of-fit tests, Journal of the American Statistical Association, 2004, 99(466): 368–382.

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang L, Brown L D, and Cai T T, A dierence based approach to semiparametric partial linear model, Electronic Journal of Statistics, 2011, 5: 619–641.

    Article  MathSciNet  MATH  Google Scholar 

  20. Zambom A Z and Akritas M G, Nonparametric lack-of-fit testing and consistent variable selection, Statistica Sinica, 2014, 24: 1837–858.

    MathSciNet  MATH  Google Scholar 

  21. Durbin J, Knott M, and Taylor C C, Components of Cramér-von Mises statistics.I, Journal of the Royal Statistical Society, Series B (Methodological), 1975, 290–307.

  22. Von Neumann J, Distribution of the ratio of the mean squared successive difference to the variance, Annals of Mathematical Statistics, 1941, 12: 367–395.

    Article  MathSciNet  MATH  Google Scholar 

  23. Harrison M J and McCabe B P M, A test for heteroscedasticity based on least squares residuals, Journal of the American Statistical Association, 1979, 74: 494–500.

    MathSciNet  MATH  Google Scholar 

  24. Brown M B and Forsythe A B, Robust tests for the equality of variances, Journal of the American Statistical Association, 1974, 69(346): 364–367.

    Article  MATH  Google Scholar 

  25. Wang L, Akritas M G, and Keilegom I V, An ANOVA-type nonparametric diagnostic test for heteroscedastic regression models, Journal of Nonparametric Statistics, 2008, 20: 365–382.

    Article  MathSciNet  MATH  Google Scholar 

  26. Chen H, Convergence rates for parametric components in a partly linear model, The Annals of Statistics, 1988, 16(1): 136–46.

    Article  MathSciNet  MATH  Google Scholar 

  27. Rice J, Convergence rates for partially splined models, Statistics & Probability Letters, 1986, 4: 203–208.

    Article  MathSciNet  MATH  Google Scholar 

  28. Speckman P, Kernel smoothing in partial linear models, Journal of the Royal Statistical Society, Series B, 1988, 50: 413–436.

    MathSciNet  MATH  Google Scholar 

  29. Green P, Jennison C, and Seheult A, Analysis of field experiments by least squares smoothing, Journal of the Royal Statistical Society, Series B, 1985, 47: 299–315.

    MathSciNet  Google Scholar 

  30. Heckman N E, Spline smoothing in partly linear models, Journal of the Royal Statistical Society, Series B, 1986, 48: 244–248.

    MathSciNet  MATH  Google Scholar 

  31. Chatterjee S B, Price, Regression Analysis by Example, Wiley, New York, 1977.

    MATH  Google Scholar 

  32. Zambom A Z and Akritas M G, Nonparametric significance testing and group variable selection, Journal of Multivariate Analysis, 2015, 133: 51–60.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, China

    Waled Khaled, Jinguan Lin & Zhongcheng Han

  2. Department of Applied Statistics, Faculty of Economics, Damascus University, Damascus, Syria

    Waled Khaled

  3. School of Statistics and Mathematics, Nanjing Audit University, Nanjing, 211815, China

    Jinguan Lin, Yanyong Zhao & Hongxia Hao

Authors
  1. Waled Khaled
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  2. Jinguan Lin
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  3. Zhongcheng Han
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  4. Yanyong Zhao
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  5. Hongxia Hao
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Corresponding author

Correspondence to Jinguan Lin.

Additional information

This work is partly supported by the National Natural Science Foundation of China under Grant Nos. 11571073, 11701286, NSF, JS (BK20171073).

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Khaled, W., Lin, J., Han, Z. et al. Test for Heteroscedasticity in Partially Linear Regression Models. J Syst Sci Complex 32, 1194–1210 (2019). https://doi.org/10.1007/s11424-019-7374-2

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  • DOI: https://doi.org/10.1007/s11424-019-7374-2

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