Summary
We attempt to investigate the effects of using residuals from robust regression replacing OLS residuals in test statistics for the normality of the errors. We have found that this can lead to substantially improved ability to detect lack of normality in suitable situations. We derive the asymptotic distribution of the robustified normality test as chi-squared with 2 degrees of freedom under the null hypothesis of normality of the error terms. The high breakdown property of the test statistic is discussed. By using simulations, we have found that situations where a small subpopulation exhibits characteristics which are different from the main population are the ones which ideally suit to the use of robustified normality tests. We have employed several real data sets from the literature to show that these types of situations arise frequently in real data sets.
* The authors would like to thank Mehmet Caner for helpful comments and suggestions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
C. Agostinelli and M. Markatou. Tests of hypotheses based on the weighted likelihood methodology. Statistica Sinica, 11(2):499–514, 2001.
A.C. Atkinson. Masking unmasked. Biometrika, 73:533–554, 1986.
R. Davidson and J. MacKinnon. Estimation and inference in econometrics. 568–571. Oxford Univ. Press, New York, 1993.
F.Y. Edgeworth. On observations relating to several quantities. Hermathena, 6:279–285, 1887.
X. He, D.G. Simpson, and S.L. Portnoy. Breakdown robustness of tests. J. Am. Statist. Assoc., 85:446–452,1990.
O. Hössier. Rank-based estimates in the linear model with high breakdown point. J. Am. Statist. Assoc., 89:149–158, 1994.
C.M. Jarque and A.K. Bera. A test for normality of observations and regression residuals. Int. Statistical Review, 55:163–172, 1987.
W.S. Krasker and R.E. Welsch. Efficient bounded influence regression estimation. J. Am. Statist. Assoc., 77:595–604, 1982.
M. Markatou, A. Basu, and B.G. Lindsay. Weighted likelihood estimating equations with a bootstrap root search. J. Am. Statist. Assoc., 93:740–750, 1998.
M. Markatou and X. He. Bounded influence and high breakdown point testing procedures in linear models. J. Am. Statist. Assoc., 89:543–549, 1994.
K. Metin. The relationship between inflation and the budget deficit in Turkey. J. of Business and Economic Statistics, 16(4):412–422, 1998.
M. Orhan and A. Zaman. Econometric applications of high-breakdown robust regression techniques. Technical report, Economics Department, Bilkent University, Ankara, 1999.
E.S. Pearson, R.B. D’Agostina, and K.O. Bowman. Tests for departure from normality: Comparison of powers. Biometrika, 64(2):231–246, 1977.
D.A. Pierce and R.J. Gray. Testing normality of errors in regression models. Biometrika, 69(1):233–236, 1982.
P.J. Rousseeuw. Least median of squares regression. J. Am. Statist. Assoc., 79:871–880, 1984.
P.J. Rousseeuw. Introduction to positive breakdown methods. In G.S. Maddala and C.R. Rao, editors, Handbook of Statistics, Vol. 15: Robust Inference, pages 101–121. Elsevier, Amsterdam, 1997.
P.J. Rousseeuw and A.M. Leroy. Robust regression and outlier detection. Wiley, New York, 1987.
P.J. Rousseeuw and K. Van Driessen. Computing LTS regression for large data sets. Technical report, University of Antwerp, 1998. Submitted.
P.J. Rousseeuw and V.J. Yohai. Robust regression by means of S-estimators. In J. Frank, W. Härdie, and R.D. Martin, editors, Robust and Nonlinear Time Series Analysis. Lecture Notes in Statistics, volume 26, pages 256–272. Springer, New York, 1984.
A. Tansel. Cigarette demand, health scares and education in Turkey. Applied Economics, 25: 521–529,1993.
C.M. Urzua. On the correct use of omnibus tests for normality. Economics Letters, 53: 247–251,1996.
H. White and G.M. Macdonald. Some large sample tests for nonnormality in the linear regression model. J. Am. Statist. Assoc., 75:16–28, 1980.
V.J. Yohai. High breakdown-point and high efficiency robust estimates for regression. Ann. Statist., 15:642–656, 1987.
V.J. Yohai and R.H. Zamar. High breakdown point estimates of regression by means of the minimization of an efficient scale. J. Am. Statist. Assoc., 83:406–413, 1988.
A. Zaman. Statistical foundations for econometric techniques. Academic Press, New York, 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Önder, A.Ö., Zaman, A. (2003). A Test for Normality Based on Robust Regression Residuals. In: Dutter, R., Filzmoser, P., Gather, U., Rousseeuw, P.J. (eds) Developments in Robust Statistics. Physica, Heidelberg. https://doi.org/10.1007/978-3-642-57338-5_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-57338-5_26
Publisher Name: Physica, Heidelberg
Print ISBN: 978-3-642-63241-9
Online ISBN: 978-3-642-57338-5
eBook Packages: Springer Book Archive