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
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.
Härdle W, Liang H, and Gao J T, Partially Linear Models, Springer-Physica-Verlag, Heidelberg, 2000.
Greene W H, Econometric Analysis, 7th Edition, Prentice Hall, New Jersey, 2011.
Ruppert D, Wand M P, and Carroll R J, Semiparametric Regression, Cambridge University Press, Cambridge, 2003.
Stone J C, Additive regression and other nonparametric models, Annals of Statistics, 1985, 13: 689–705.
Hastie T J and Tibshirani R J, Generalized Additive Models, Monographs on Statistics and Applied Probability, Chapman and Hall, London, 1990.
Härdle W, Mori Y, and Vieu P, Statistical Methods for Biostatistics and Related Fields, Berlin, 2006.
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.
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.
Wang L and Zhou X H, A fully nonparametric diagnostic test for homogeneity of variances, Canadian Journal of Statistics, 2005, 33: 545–558.
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.
Diblasi A and Bowman A, Testing for constant variance in a linear model, Statistics & Probability Letters, 1997, 33(1): 95–103.
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.
Francisco-Fernndez M and Vilar-Fernndez J M, Two tests for heteroscedasticity in nonparametric regression, Computational Statistics, 2009, 24(1): 145–163.
Zheng X, Testing heteroscedasticity in nonlinear and nonparametric regressions, The Canadian Journal of Statistics, 2009, 37: 282–300.
Dette H and Marchlewski M, A robust test for homoscedasticity in nonparametric regression, Journal of Nonparametric Statistics, 2010, 22(6): 723–36.
Levene H, Robust tests for equality of variances, Probability and Statistics, Essays in Honor of Harold Hotelling, Stanford University Press, Stanford, 1960, 278–292.
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.
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.
Zambom A Z and Akritas M G, Nonparametric lack-of-fit testing and consistent variable selection, Statistica Sinica, 2014, 24: 1837–858.
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.
Von Neumann J, Distribution of the ratio of the mean squared successive difference to the variance, Annals of Mathematical Statistics, 1941, 12: 367–395.
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.
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.
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.
Chen H, Convergence rates for parametric components in a partly linear model, The Annals of Statistics, 1988, 16(1): 136–46.
Rice J, Convergence rates for partially splined models, Statistics & Probability Letters, 1986, 4: 203–208.
Speckman P, Kernel smoothing in partial linear models, Journal of the Royal Statistical Society, Series B, 1988, 50: 413–436.
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.
Heckman N E, Spline smoothing in partly linear models, Journal of the Royal Statistical Society, Series B, 1986, 48: 244–248.
Chatterjee S B, Price, Regression Analysis by Example, Wiley, New York, 1977.
Zambom A Z and Akritas M G, Nonparametric significance testing and group variable selection, Journal of Multivariate Analysis, 2015, 133: 51–60.
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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